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TWENTIETH   CENTURY  TEXT-BOOKS 


LORD    KELVIN    (SIR    WILLIAM    THOMSON)    (1824-19GT) 

William  Thomson  ranks  as  one  of  the  two  or  three  greatest 
physicists  of  the  nineteenth  century.  He  was  born  in  Ireland,  but 
spent  nearly  his  entire  life  in  Glasgow,  Scotland,  where  his  father 
was  professor  of  mathematics.  At  the  early  age  of  seventeen  he 
began  to  write  on  mathematical  subjects,  but  his  attention  was  soon 
turned  to  the  study  of  physics.  In  1846  he  became  professor  of 
natural  philosophy  at  Glasgow,  where  he  remained  fifty-three  years. 

Thomson's  early  investigations  led  to  the  invention  of  the  abso- 
lute scale  of  temperature.  Important  experiments  in  heat  were 
carried  out  by  him  in  collaboration  with  Joule  from  1852-1862. 

Later  Thomson  became  almost  universally  known  by  his  work 
as  electrician  of  the  Atlantic  cables,  the  first  of  which  was  laid  in 
1858.  By  the  invention  of  the  well-known  mirror  galvanometer 
which  he  used  in  receiving  messages,  he  increased  the  rate  of  trans- 
mission from  two  or  three  to  about  twenty-five  words  per  minute. 
This  instrument  was  replaced  later  by  his  "  siphon  recorder,"  which 
is  still  in  use. 

Thomson  was  instrumental  in  establishing  the  practical  system  of 
electrical  units.  He  invented  the  absolute  and  quadrant  electrome- 
ters for  measuring  potentials,  a  sounding  device  for  use  on  moving 
ships,  and  many  other  practical  measuring  instruments.  His  favorite 
subject,  however,  was  the  nature  of  the  ether,  which,  as  he  said, 
claimed  his  attention  daily  for  over  forty  years.  His  writings  are 
included  in  his  Papers  on  Electrostatics  and  Magnetism,  Mathemat- 
ical and  Physical  Papers,  Popular  Lectures  and  Addresses,  and 
Thomson  and  Tait's  An  Elementary  Treatise  on  Natural  Philosophy. 


TWENTIETH   CENTURY    TEXT-BOOKS 


A  HIGH  SCHOOL  COURSE 
IN  PHYSICS 


BY 


FREDERICK   R.   GORTON,   B.S.,   M.A.,  PH.D. 

ASSOCIATE  PROFESSOR  OF   PHYSICS,    MICHIGAN 
STATE  NORMAL  COLLEGE 


or -HE 
UNIVERSITY 


D.     APPLETON    AND     COMPANY 

NEW    YORK  CHICAGO 

1910 


COPYRIGHT,  1910,  BY 
D.   APPLETON  AND  COMPANY. 


oe. 


PREFACE 

PHYSICS  is  the  summary  of  a  part  of  human  experience.  Its  devel- 
opment has  resulted  from  the  fact  that  its  pursuit  has  successfully 
met  human  needs.  Hence  it  is  believed  that  the  presentation  of  the 
subject  in  the  secondary  school  should  be  the  expansion  of  the  every- 
day life  of  the  pupil  into  the  broader  experience  and  observation  of 
those  whose  lives  have  been  devoted  to  the  study.  Human  activity 
and  progress,  therefore,  should  be  the  teacher's  guiding  principle, 
and  the  bearing  of  each  phenomenon  and  law  on  the  interests  of 
mankind  should  be  clearly  disclosed  and  emphasized.  The  author 
of  this  book  has  accordingly  endeavored  to  give  great  prominence 
to  facts  of  common  observation  in  the  derivation  of  physical  laws, 
and,  further,  has  attempted  to  point  out  plainly  the  service  that  has 
been  afforded  mankind  by  a  knowledge  of  nature's  laws.  No  text- 
book, however,  can  take  the  place  of  the  skilled  teacher  in  showing 
clearly  the  relation  of  physical  phenomena  to  human  activities,  or  in 
the  selection  of  illustrative  examples  within  the  range  of  observation 
of  his  pupils. 

Large  portions  of  the  subject-matter  of  Physics  deal  with  knowl- 
edge already  possessed  by  a  pupil  of  high-school  age,  and  nothing 
is  of  more  appealing  interest  to  him  than  the  feeling  that  this  infor- 
mation is  to  be  made  of  some  value.  By  recalling  phenomena  well 
within  the  acquaintance  of  the  pupil  and  supplementing  them  with 
demonstrative  experiments,  the  way  is  easily  paved  to  the  deduction 
and  interpretation  of  general  principles.  In  the  presentation  of  such 
experiments,  the  author  has  described  as  simple  and  inexpensive 
apparatus  as  he  has  found  to  be  consistent  with  satisfactory  results. 

No  effort  has  been  spared  to  give  the  teacher  and  pupil  every  pos- 
sible assistance.  The  sections  have  been  plainly  set  off  and  given 
suggestive  headings  ;  references  to  related  material  have  been  inserted 
where  needed;  and  the  numerous  sets  of  exercises  have  been  care- 

v 

209710 


yi  PREFACE 

fully  graded.  In  order  to  bring  the  problems  near  the  discussions 
upon  which  their  solutions  depend,  they  have  been  arranged  in  more 
and  smaller  groups  than  is  usual.  The  exercises  throughout  the 
book  have  been  selected  from  concrete  cases,  and  the  usual  problems 
in  pure  reductions  have  been  omitted.  Illustrative  solutions  of  prob- 
lems and  suggestions  have  been  given  wherever  difficulties  have  been 
found  to  arise.  As  an  aid  to  the  pupil  in  reviewing  and  to  the 
teacher  in  conducting  rapid  drill  exercises,  a  summary  of  the  contents 
of  each  chapter  is  presented  at  its  conclusion. 

The  educational  value  of  the  portraits  and  biographical  sketches 
of  many  of  the  great  men  of  science  is  at  once  apparent.  Emphasis 
upon  the  parts  that  these  men  have  played  in  the  development  of 
Physics  has  been  recognized  by  eminent  educators  as  an  important 
factor  in  creating  an  atmosphere  of  human  interest  around  the  sub- 
ject. These  names  should  become  familiar  to  every  student  of  Physics. 

On  account  of  the  rapid  advance  at  the  present  time  in  the  practi- 
cal uses  made  of  physical  principles,  the  author  believes  that  the 
sections  involving  new  applications  will  be  found  of  general  interest 
and  utility. 

The  book  will  be  found  to  be  free  from  the  more  difficult  uses 
of  algebraic  and  geometric  principles.  The  place  of  first  importance 
has  been  given  to  the  study  of  phenomena,  and  mathematical  expres- 
sions have  been  introduced  as  convenient  means  of  designating  exact 
relations  which  have  been  previously  interpreted.  It  is  mainly  in  the 
subject  of  Physics  that  the  pupil  is  brought  to  realize  the  value 
of  his  mathematical  studies  in  the  world  of  concrete  quantities. 

The  author  believes  that  the  class-room  work  in  the  subject  should 
be  accompanied  by  a  sufficient  number  of  individual  laboratory  exer- 
cises to  fix  clearly  in  mind  great  principles  and  important  phenomena. 
Further,  enough  practice  with  simple  drawing  instruments  should 
be  given  to  enforce  the  use  of  the  simplest  geometrical  relations. 

In  addition  to  the  many  subjects  whose  treatment  is  demanded  by 
the  achievements  of  recent  times,  the  author  has  given  careful  atten- 
tion to  the  various  topics  recommended  by  the  Committee  of  Second- 
ary School  Teachers  to  the  College  Entrance  Examination  Board. 

The  author  desires  to  acknowledge  here  the  many  helpful  sugges- 
tions of  those  who  have  read  and  criticised  the  proof  or  manuscript,  and 


PREFACE  vii 

wishes  especially  to  mention  Professor  E.  A.  Strong  of  the  Michigan 
State  Normal  College ;  Mr.  Fred  R.  Nichols  of  the  Richard  T.  Crane 
Manual  Training  High  School,  Chicago,  111. ;  Mr.  Frank  B.  Spaulding 
of  the  Boys'  High  School,  Brooklyn ;  Mr.  Albert  B.  Kimball,  Princi- 
pal of  the  High  School,  Fairhaven,  Mass. ;  Professor  Karl  E.  Guthe 
of  the  University  of  Michigan;  Mr.  George  A.  Chamberlain,  Prin- 
cipal of  East  Division  High  School,  Milwaukee;  and  Mr.  J.  M. 
Jameson  of  the  Pratt  Institute,  Brooklyn,  N.Y. 


CONTENTS 

CHAPTER   I 

PAGE 

/  INTRODUCTION 1 

CHAPTER   II 

MOTION,  VELOCITY,  AND  ACCELERATION 

1.  Uniform  Motions  and  Velocities 13 

2.  Uniformly  Accelerated  Motion 16 

3.  Simultaneous  Motions  and  Velocities       .        .  .21 
Summary     ......  26 

CHAPTER   III 

LAWS  or  MOTION  —  FORCE 

1.  Discussion  of  Newton's  Laws 28 

2.  Concurrent  Forces 35 

3.  Moments  of  Force  —  Parallel  Forces         ....  39 

4.  Resolution  of  Forces 42 

5.  Curvilinear  Motion    ........  44 

Summary     ..........  48 

CHAPTER   IV 

WORK  AND  ENERGY 

1.  Definition  and  Units  of  Work 51 

2.  Activity,  or  Rate  of  Work 53 

3.  Potential  and  Kinetic  Energy  .        .  .        .        .54 

4.  Transitions  of  Energy 58 

Summary     ..........       61 

CHAPTER   V 
GRAVITATION 

1.  Laws  of  Gravitation  and  Weight 62 

2.  Equilibrium  and  Stability 65 

ix 


X  CONTENTS 

PAGK 

3.  The  Fall  of  Unsupported  Bodies   ...        .        .        .69 

4.  The  Pendulum 74 

Summary     .        .         .        .        .        •         .         •         •         .81 

CHAPTER  VI 
MACHINES 

1.  General  Law  and  Purpose  of  Machines    ....      83 

2.  The  Principle  of  the  Pulley      .        .        .        .        .        .85 

3.  The  Principle  of  the  Lever 88 

4.  The  Principle  of  the  Wheel  and  Axle       .        .       „        .93 

5.  The  Inclined  Plane,  Screw,  and  Wedge    .         .        .        .      96 

6.  Efficiency  of  a  Machine      .  -     .        •        .        .        •        •      99 
Summary 101 

CHAPTER  VII 
MECHANICS  OF  LIQUIDS 

1.  Forces  Due  to  the  Weight  of  a  Liquid      .        .        .        .103 

2.  Force  Transmitted  by  a  Liquid         .        .        .       •«    .     •  Ill 

3.  Archimedes'  Principle       . 119 

4.  Density  of  Solids  and  Liquids  .        .        .        .        .        .  124 

5.  Molecular  Forces  in  Liquids      .        .        .        .        .        .  130 

Summary     ..........  135 

CHAPTER  VIII 
MECHANICS  OF  GASES 

1.  Properties  of  Gases    .        .  .        .        .        .        .  138 

2.  Pressure  of  the  Air  against  Surfaces          ....  139 

3.  Expansibility  and  Compressibility  of  Gases     .        .        .  147 

4.  Atmospheric  Density  and  Buoyancy          ....  153 

5.  Applications  of  Air  Pressure     ......  155 

Summary     .         .        .        .        .        .         .         .         .        .  163 

CHAPTER  IX 

SOUND:  ITS  NATURE  AND  PROPAGATION 

1.  Origin  and  Transmission  of  Sound 165 

2.  Nature  of  Sound 169 

3.  Intensity  of  Sound 173 

4.  Reflection  of  Sound 176 

Summary    ..........  177 


CONTENTS  xi 


CHAPTER   X 

PAGE 

SOUND:  WAVE  FREQUENCY  AND  WAVE  FORM 

1.  Pitch  of  Tones 179 

2.  Resonance .        .        .  185 

3.  Wave  Interference  and  Beats    .         .        *        .        .        .188 

4.  The  Vibration  of  Strings 192 

5.  Quality  of  Sounds 196 

6.  Vibrating  Air  Columns 198 

Summary     ..........  205 

CHAPTER   XI 

HEAT  :  TEMPERATURE  CHANGES  AND  HEAT  MEASUREMENT 

1.  Temperature  and  its  Measurement   .....     208 

2.  Expansion  of  Bodies .        .216 

3.  Calorirnetry,  or  the  Measurement  of  Heat        .         .        .     225 
Summary    ..........     229 

CHAPTER,  XII 

HEAT  :  TRANSFERENCE  AND  TRANSFORMATION  OF  HEAT  ENERGY 

1.  Change  of  the  Molecular  State  of  Matter          .         .        .     231 

2.  The  Transference  of  Heat 247 

3.  Relation  between  Heat  and  Work 256 

Summary    .        .         .  .        .        .        .        .        .    265 

CHAPTER  XIII 

LIGHT  :  ITS  CHARACTERISTICS  AND  MEASUREMENT 

1.  Nature  and  Propagation  of  Light 268 

2.  Rectilinear  Propagation  of  Light 270 

3.  Intensity  and  Candle  Power  of  Lights      ....  274 
Summary 277 

CHAPTER   XIV 

LIGHT  :  REFLECTION  AND  REFRACTION 

1.  Reflection  of  Light 278 

2.  Reflection  by  Curved  Mirrors 284 

3.  Refraction  of  Light 292 


xii  CONTENTS 

PAGE 

4.  Lenses  and  Images 300 

5.  Optical  Instruments 311 

Summary 318 

CHAPTER  XV 

LIGHT:  COLOR  AND  SPECTRA 

1.  Dispersion  of  Light :  Color 321 

2.  Spectra 324 

3.  Interference  of  Light .  330 

Summary    .         .         .        .                 •        .        .         •         •  333 

CHAPTER   XVI 
ELECTROSTATICS 

1.  Electrification  and  Electrical  Charges       ....  335 

2.  Electric  Fields  and  Electrostatic  Induction      .        .        .  340 

3.  Potential  Difference  and  Capacity 346 

4.  Electrical  Generators 352 

Summary    ..........  356 

CHAPTER  XVII 
MAGNETISM 

1.  Magnets  and  their  Mutual  Action    .        .        .        •  I     •  359 

2.  Magnetism  a  Molecular  Phenomenon       .        .        •        .  365 

3.  Terrestrial  Magnetism       .......  368 

Summary    .        .        .»        «        .        .        .        .        .        .  372 

CHAPTER   XVIII 
VOLTAIC  ELECTRICITY 

1.  Production  of  a  Current  —  Voltaic  Cells  ....  374 

2.  Effects  of  Electric  Currents       .         .         .         .         .         .  388 

Summary    ..........  400 

CHAPTER   XIX 

ELECTRICAL  MEASUREMENTS 

1.  Electrical  Quantities 'and  Units        .....  402 

2.  Electrical  Energy  and  Power    ......  413 

3.  Computation  and  Measurement  of  Resistances         .         .  417 
Summary    .         .        .         .         .         .         .         .         .         .  422 


CONTENTS  xiii 


CHAPTER  XX 

PAGE 

ELECTRO-MAGNETIC  INDUCTION 

1.  Induced  Currents  of  Electricity 425 

2.  Dynamo-Electric  Machinery 432 

3.  Transformation  of  Power  and  Its  Applications        .        .  442 

4.  The  Telegraph  and  the  Telephone 453 

Summary 460 

CHAPTER  XXI 
RADIATIONS 

1.  Electro-magnetic  waves 462 

2.  Conduction  of  Gases          .......  466 

3.  Radio-activity 468 

INDEX                                                                                                  ,  473 


LIST   OF   PORTRAITS 

Lord  Kelvin  (Sir  William  Thomson)  .        .        .       Frontispiece 

FACING  PAGE 

Sir  Isaac  Newton 30 

Galileo  Galilei 70 

Hermann  von  Helmholtz      .         .         .         .         .        .        .         .  196 

Count  Rumford  (Sir  Benjamin  Thompson)          ....  256 

James  Prescott  Joule 258 

Benjamin  Franklin 346 

Count  Alessandro  Volta         .        .        .        .        .        .        .         .352 

Hans  Christian  Oersted          ........  382 

Dominique  Francois  Jean  Arago           ......  382 

Andre  Marie  Ampere .        .        .  406 

George  Simon  Ohm 406 

Michael  Faraday 426 

Joseph  Henry 426 

James  Clerk-Maxwell 432 

Heinrich  Hertz 464 

Sir  William  Crookes 466 

Wilhelm  Konrad  Rontgen     .        .        .        .  .        .        .466 

Antoine  Henri  Becquerel       .        .        .        •        .        .        .        .  470 

Madame  Curie       .        .........  470 


CHAPTER  I 
INTRODUCTION 

iX 

1.  Physics.  —  In  the  experiences  of  everyday  life  we 
witness  a  great  variety  of  changes  in  the  things  around  us. 
Objects  are  moved,  melted,  evaporated,  solidified,  bent, 
made^hot  or  cold,  and  undergo  a  change  in  their  condi- 
tion, place,  or  shape  in  a  great  many  other  ways.    Physics 
is  the  science  that  treats  of  the  properties  of  different  sub- 
stances and  the  changes  that  may  take  place  within  or 
between  bodies,  and  it  investigates  the  conditions  under 
which  such  changes  occur. 

yin  its  broadest  sense  Physics  is  the  science  of  phenom- 
ena. Every  action  of  which  we  become  aware  through 
the  senses  is  a  phenomenon.  We  hear  the  rolling  thunder, 
we  see  the  shining  of  a  live  coal,  we  taste  the  dissolving 
sugar,  we  smell  the  evaporating  oil,  and  we  feel  moving 
air.  By  considering  his  own  experience  the  student  will 
be  able  to  recall  numerous  examples  of  physical  phe- 
nomena and  to  state  the  sense  by  means  of  which  he  per- 
ceives each  of  them. 

>^  The  study  of  physics,  however,  not  only  directs  our 
attention  to  the  phenomena  to  which  we  are  accustomed, 
but  to  a  multitude  of  m'ore  unusual  but  not  less  important 
ones.  It  also  strives  to  put  these  phenomena  to  experi- 
mental tests  that  will  enable  us  to  understand  the  laws 
connecting  actions  with  their  causes. 

2.  Utility  of  the  Study  of  Physics.  —  Increasing  acquaint- 
ance with  nature  and  naturaj_Jaw  has  been  the  means  of 

2  ~T" 


2  A  HIGH   SCHOOL   COURSE  IN  PHYSICS 

elevating  man  from  the  life_of  limited  power  and  useful- 
ness of  the  savage  to  his  condition  of  present-day  enlight- 
enment. The  early  discovery  of  fire  was  a  great  step 
toward  civilization.  By  some  crude  experimental  study  it 
was  found  later  that  fire  could  be  produced  at  will,  as  by 
the  striking  of  flint  and  by  rubbing  two  pieces  of  dry  wood 
together.  Thus  the  observation  of  simple  natural  phe- 
nomena enabled  man  to  secure  heat  for  cooking  his  food 
and  warming  his  habitation,  besides  aiding  him  in  forming 
implements  to  procure  food,  improve  his  shelter,  and  give 
him  protection  from  enemies. 

This  same  observation  of  natural  phenomena  has  pro- 
duced every  existing  artificial  device  for  our  protection, 
convenience,  and  comfort.  The  engineer  who  plans  a  rail- 
road, with  its  bridges,  tunnels,  and  grades,  together  with 
the  locomotive  and  its  train,  makes  use  at  every  step  of 
knowledge  acquired  through  the  study  of  Physics.  The 
surveyor  ascertains  how  to  cut  through  the  hills  and  fill 
the  valleys  by  the  use  of  instruments  which  involve  physi- 
cal principles.  By  the  discovery  and  application  of  physi- 
cal laws  scientists  and  inventors  have  produced  the  tel- 
escope, telephone,  steam  engine,  electric  car,  and  all  the 
other  useful  appliances  which  form  so  important  a  part  of 
our  everyday  life. 

3.  Matter.  —  There  are  three  general  characteristics  by 
which  matter  is  recognized. 

(1)  Matter  always  occupies  space.  On  this  account  it 
is  said  to  possess  the  property  of  extension.  Many  invisi- 
ble bodies  of  matter  exist.  The  air  in  a  bottle  or  tumbler 
is  such  a  body.  We  may  show,  however,  by  the  following 
experiment  that  it  is  as  real  as  any  other  body: 

1.  Place  a  piece  of  cork  upon  the  surface  of  water  in  a  vessel,  cover 
it  with  an  inverted  tumbler,  and  force  the  tumbler  deep  into  the 
water  as  in  Fig.  1.  The  air  that  was  in  the  tumbler  still  occupies 


INTRODUCTION  3 

nearly  all  of  that  space,  and  the  water  is  not  allowed  to  rise  and 
fill  it. 

(2)  All  matter  is  indestructible,  in  the  sense  that  it  has 
never  been  discovered  that  the  small- 
est portion  can  be  annihilated  by  any 

process  known  to  man.  Causing  a 
body,  as  a  piece  of  coal,  to  disappear 
by  burning  does  not  destroy  the  ma- 
terial of  which  it  is  composed.  A 
portion  is  carried  away  in  the  smoke 
and  gases,  and  the  remainder  left  be-  Fia-  i.  — inverted  Tumbler 

.  .     ,& .        '  Nearly  Full  of  Air. 

hind  in  the  ash  produced.     In  the 

process  of  evaporation  a  drop  of  water  becomes  invisible ; 
but  the  matter  still  exists  in  the  atmosphere  as  a  trans- 
parent vapor. 

(3)  All   matter   has   weight,  i.e.    is   attracted   by  the 
earth.     In  a  more  general  sense  it  may  even  be  said  that 
every  body  has   an   attraction  for,  or  pulls  upon,  every 
other  body.     The  term  gravitation  is  used  to  express  this 
characteristic  of  matter. 

The  fact  that  air  possesses  weight  as  well  as  extension 
may  be  shown  by  experiment  as  follows : 

2.  Remove  the  brass  fixture  from  an  incandescent  lamp  bulb,  and 
carefully  balance  the  bulb  on  a  sensitive  beam  balance.  Introduce  a 
short  nail  into  the  stem  of  the  bulb,  and  tap  lightly  with  a  hammer 
until  the  glass  is  broken  and  air  admitted.  If  the  bulb  is  now  placed 
upon  the  balance,  a  decided  increase  in  weight  will  be  observed. 
The  weight  of  air  admitted  has  been  added  to  that  of  the  bulb,  which 
originally  contained  almost  no  air. 

The  quantity  of  matter  in  a  body  is  called  its  mass.    Dif-    j 
ferent  kinds  of  matter,  as  gold,  water,  glass,  air,  mercury,    < 
salt,  hydrogen,  etc.,  are  called  substances.     Substances  are 
recognized  by  their  properties;    as,  hardness,  elasticity, 
tenacity,  fluidity,  transparency,  etc. 


4  A  HIGH   SCHOOL   COURSE   IN   PHYSICS 

4.  Measurement  of  Quantities.  —  A  little  reflection  will 
show  the  necessity  of  having  systems  of  measurement  for 
the  various  quantities  that  we  find  in  nature,  such  as 
length,  area,  volume,  weight,  time,  etc.     The  importance 
of  such  systems  has  been  recognized  by  the  governments 
of  civilized  countries,  and  the  values  of  the  units  employed 
have  been  fixed  by  law.     Thus  the  foot,  pound,  second, 
etc.,  are  well-established  units  of  quantity,  and  are  in  gen- 
eral use  throughout  Great  Britain  and  the  United  States. 

The  English  system  of  measurement,  however,  is  objec- 
tionable on  account  of  the  inconvenient  relations  between 
the  units  and  their  multiples  and  divisions.  For  example, 
1  pound  =  16  ounces  =  7000  grains ;  or  1  mile  =  320  rods 
=  5280  feet  =  63,360  inches.  It  is  mainly  for  this  reason 
that  many  countries  have  adopted  the  metric  system  of 
measurement,  in  which  the  relations  are  always  to  be 
expressed  by  some  power  of  ten.  This  system  greatly  re- 
duces the  effort  required  in  making  correct  computations.1 
Since  in  the  United  States  both  the  English  and  the 
metric  system  are  employed,  it  will  be  advisable  to  be- 
come proficient  in  the  use  of  each. 

5.  Measures  of  Extension.  —  Every  body  occupies  space 
of  three  dimensions :  length,  breadth,  and  thickness ;  but 
each  of  these  is  simply  a  length,  the  metric  unit  of  which 
is  the  meter.     The  meter  is  the   distance   between  two 
transverse  lines  ruled  on  a  platinum-iridium  bar  kept  in 
thje  Archives  at  Sevres,  near  Paris.2     On  account  of  the 

1  Congress  recognized  the  desirability  of  introducing  the  metric  system 
as  early  as  1866.     See  Congressional  Globe,  Appendix,  Part  5,  p.  422, 
Chap.  CCCI :   An  act  to  authorize  the  use  of  the  metric  system  of  weights 
and  measures.     By  this  act  the  yard  is  defined  as  f ff$  of  a  meter. 

2  The  meter  was  originally  intended  to  be  one  ten-millionth  of  the  dis- 
tance from  the  equator  to  the  north  pole.     Accurate  copies  of  the  meter 
and  other  metric  units  are  kept  in  the  U.  S.  Bureau  of  Standards  at 
Washington,  D.C.. 


INTRODUCTION  5 

changes  in  the  length  of  the  bar  with  variations  of  tem- 
perature, the  distance  must  be  taken  when  the  bar  is  at 
the  temperature  of  freezing  water.  The  multiples  and 
divisions  of  the  meter  are  designated  by  prefixes  signify- 
ing the  relations  which  they  bear  to  the  unit.  The  multi-  . 
pies  have  the  Greek  prefixes,  deka  (ten),  hecto  (hundred), 
kilo  (thousand),  and  myria  (ten  thousand).  The  divisions 
have  the  Latin  prefixes,  deci  (tenth),  centi  (hundredth), 
and  milli  (thousandth).  The  relations  are  shown  in  the 
following  table  : 

METRIC  TABLE  OF  LENGTH 
1  A  myriameter  equals  10,000  meters. 

A  kilometer      (km.)  equals    1,000  meters. 
1  A  hectometer  equals       100  meters. 

1 A  dekameter  equals         10  meters. 

A  decimeter  1  (dm.)  equals          0.1      of  a  meter. 

A  centimeter     (cm.)  equals          0.01    of  a  meter. 

A  millimeter   (mm.)  equals          0.001  pf  a  meter. 

The  most  important  equivalents  in  the  English  system 
are  the  following : 

1  meter          (m.)  equals  39.37      inches,  or  1.094  yards. 
2  1  centimeter  equals    0.3937  of  andnch. 

1  kilometer  equals    0.6214  of  a  mile. 

The  metric  equivalents  most  frequently  used  are : 

1  yard  equals  91.44  centimeters,  or  0.9144  m. 
2  1  inch  equals    2.540  centimeters. 
1  mile  equals    1.609  kilometers. 

The  relative  sizes  of   the   inch   and  the  centimeter  are 
shown  in  Fig.  2. 

Because  the  meter  is  too  large  for  general  use  in  Physics, 

the  centimeter  has  been  chosen  as  the  unit.     The  centimeter 

and  the  gram,  which  is  the  metric  unit  of  mass,  and  the 

second  as  a  time  unit,  are  together  the  fundamental  units 

1  Seldom  used.  2  To  be  memorized. 


A   HIGH   SCHOOL  COURSE   IN   PHYSICS 


Fia.  3. —  Relative 
Sizes  of  the  Square 
Inch  and  Square 
Centimeter. 


of  the  so-called  centimeter -gram- second  (C.  G.  S.)  system 
of  measurement  which  is  in  use  throughout  the  world  in 

scientific  work. 

s  6.  Surface  Measure.  —  The  unit  of  sur- 
— L-  ^  face,  or  area,  in  the  C. 
G.  S.  system  is  the  square 
centimeter  (cm.2).  It  is 
the  area  of  a  square  whose 
edge  is  one  centimeter  in 
length.  One  square 
~ OT  inch  is  thus  obviously 
equal  to  (2.540)2,  or 
6.4516  square  centi- 
meters. The  relative 
sizes  of  these  two  units  are  shown  in  Fig.  3. 
7.  Cubic  Measure.  —  The  unit  of  volume 
in  the  C.  G.  S.  system 
is  the  cubic  centimeter 
(cm.3).  This  unit  is 
defined  as  the  vol- 
ume of  a  cube  whose 

^   edge    is     one    centi- 

meter  in  length. 
One  cubic  inch  thus 
equals  (2.540)3,  or 
16.387  cubic  centi- 
— »  meters.  The  relative 

—  o   sizes  of  these  units  are  shown  in  Fig.  4. 
FIG.   2.  —  Showing      g.    Measures  of  Capacity. — The  unit  of 

the  Relative  Sizes  ._,  ,-•        7.. 

of    the    English  capacity  in  the  metric  system  is  the  liter 
inch  and  the  Met-  (pronounced  lee'ter*),    which    is   equal   in 

ric  Centimeter.         .  ,  .       ,      .  .•*  -, 

size  to  a  cubic  decimeter,  or  one  thousand 
cubic  centimeters.  The  liter  is  somewhat  larger  than  the 
liquid  quart  and  smaller  than- the  dry  quart.  More  pre- 


CO 


<N 


FIG.  4.  —  Relative  Sizes 
of  the  Cubic  Inch  and 
Cubic  Centimeter. 


INTRODUCTION  .  7 

cisely,  a  liter  equals  1.057  liquid  quarts  and  0.908  of  a 
dry  quart.  Multiples  and  divisions  of  the  liter  are  des- 
ignated by  the  prefixes  explained  in  §  5,  but  are  little 
used  in  ordinary  physical  measurements. 

EXERCISES 

1.  The  distance  from  Detroit  to  Chicago  is  280  mi.     What  is  the 
metric  equivalent  of  this  distance  ? 

2.  Which  is  the  lower  price  for  silk,  $1  per  yard  or  $1.10  per  meter  ? 
How  much  is  the  difference  ? 

3.  A  tourist  while  in  Paris  pays  the  equivalent  of  50  ct.  per  meter 
for  cloth  worth  40  ct.  per  yard.     How  much  is  the  loss  on  a  purchase 
of  20  m.? 

4.  If  the  cost  of  water  is  10  ct.  per  thousand  gallons,  what  is  the 
equivalent  cost  per  cubic  meter?     (1  gal.  =  231  cu.  in.) 

5.  How  much  dearer  in  Germany  is  oil  costing  the  equivalent  of 
5  ct.  per  liter  than  the  same  in  the  United  States  at  15  ct.  per  gallon? 
Express  the  result  in  cents  per  gallon. 

6.  How  much  more  cloth  will  $1  buy  at  20  ct.  per  meter  than 
at  18  ct.  per  yard  when  the  width  is  30  in.  ?    Express  the  result  in 
square  inches. 

7.  If  a  railroad  ticket  in  France  costs  the  equivalent  of  $19.50  per 
thousand  kilometers,  what  is  the  rate  per  mile? 

8.  If  illuminating  gas  in  Germany  is  sold  at  the  equivalent  of 
3.5  ct.  per  cubic  meter,  what  is  the  corresponding  price  per  thousand 
cubic  feet  ? 

9.  Measures  of  Mass.  —  The  unit  of  mass  in  the  metric 
system  is  the  kilogram,  and  in  the  C.  G.  S.  system,  the 
gram.     The  gram-mass  is  the  one-thousandth  part  of  the 
mass  of  a  standard  platinum-iridium  cylinder  preserved  in 
the  Archives  of  France.     The  entire  mass  of  this  cylinder 
is  one  kilogram  (abbreviated  to  kilo,  pronounced  kee'lo). 
This  standard  kilogram  was  intended  to  be  equal  to  the 
mass  of  one  thousand  cubic  centimeters,  or  one  liter,  of 
pure  water;  in  fact,  it  may  be  considered  so  without  ap- 
preciable error. 

This  relation  between  mass  and  volume  in  the  metric 


8  A  HIGH    SCHOOL  COURSE   IN  PHYSICS 

system  is  of  great  convenience   in   physics.     Since   one 
cubic  centimeter  of  water  has  a  mass  of  one  gram,  if  we 


FIG.  5.  — Relation  of  the  Unit  of  Mass  to  the  Unit  of  Volume. 

know  the  mass  in  grams  of  a  certain  volume  of  water,  the 
volume  also  is  known,  and  vice  versa.     (See  Fig.  5.) 

METRIC  TABLE  OF  MASS 
1  A  myriagram  equals  10,000  grams  (g.). 

A  kilogram       (kg.)  equals    1,000  grams. 
1  A  hectogram  equals      100  grams. 

1 A  dekagram  equals         10  grams. 

A  decigram      (dg.)  equals          0.1      of  a  gram. 

A  centigram     (eg.)  equals          0.01     of  a  gram. 

A  milligram    (mg.)  equals          0.001  of  a  gram. 

The  English  unit  of  mass  used  in  Physics  is  the  avoir- 
dupois pound  containing  7000  grains  and  denned  as  being 
2~23T6  °f  a  kilogram.  Its  multiple  is  the  ton,  or  2000 
pounds ;  its  divisions,  the  ounce  and  the  grain. 

The  English  and  metric  equivalents  most  frequently 
used  are  as  follows  : 

1  pound       is  equal  to  453.59  grams. 

1  ounce        is  equal  to    28.35  grams. 

2  1  kilogram  is  equal  to      2.20  pounds. 

1  gram         is  equal  to    15.43  grains. 

1  Seldom  used.  2  To  be  memorized. 


INTRODUCTION  9 

EXERCISES 

1.  Express  the  mass  of  a  cubic  inch  of  water  in  grams. 

2.  What  is  the  mass  of  a  cubic  decimeter  of  water  in  pounds  ? 

3.  Sugar  at  6  ct.  per  pound  costs  how  much  per  kilogram  ? 

4.  A  cubic  centimeter  of  mercury  has  a  mass  of  13.6  g.     Find  the 
mass  of  a  cubic  inch  of  mercury  in  ounces.  Ans.  7.86  oz. 

5.  How  many  pounds  are  there  in  a  cubic  foot  of  water? 

6.  How  many  grams  of  water  will  be  required  to  fill  a  rectangular 
vessel  measuring  20  x  25  x  30  cm.  ?     Reduce  to  pounds. 

7.  One  cubic  centimeter  of  iron  has  a  mass  of  7.5  g.     Find  the 
mass  of  an  iron  plate  150  cm.  square  and  2  mm.  thick. 

8.  An  empty  flask  weighs  100  g. ;  when  filled  with  water,  the  entire 
mass  is  365  g.     What  is  the  capacity  of  the  flask  ? 

9.  If  the  mass  of  a  given  volume  of  gold  is  19  times  that  of  an 
equal  volume  of  water,  what  is  the  mass  of  25  cm.3  of  gold?    What  is 
the  volume  of  a  gold  body  whose  mass  is  10  g.  ? 

10.  Mass  Distinguished  from  Weight. — Mass  and  weight 
must  not  be  regarded  as  synonymous  terms.     If  it  were 
not  for  the  fact  that  any  two  masses  attract  each  other 
(§  3),  we   should   have  little  use   for   the  word   weight. 
This  attraction  between  common  bodies  is  beyond  our 
power  to  detect  by  ordinary  means,  because  it  is  so  slight. 
When,  however,  one  of  the  attracting  bodies  is  massive, 
as  the  earth,  and  the  other  is  some  object,  as  a  stone,  the 
attraction  is  great  enough  to  be  easily  perceived  as  we 
try  to  support  or  lift  the  smaller  body.      This  downward 
putt  of  the  earth  upon  the  stone  is  called  the  weight  of  the 
stone.     The  weight,  or  earth-pull,  of  a  body  changes  when 
it  is  taken  to  a  different  latitude,  or  is  elevated  above,  or 
lowered  beneath,  the  surface  of  the  earth  (§  68).     It  is 
therefore  obvious  that  the  weight  of  a  body  may  change 
while  the  quantity  of  matter  in  it,  i.e.  its  mass,  remains 
the  same. 

11.  Processes  of  Weighing.  —  Since  equal  masses  are  at- 
tracted equally  by  the  earth  at  any  given  place,  weighing 


10 


A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


FIG.  6.  —  A  Dyna- 
mometer or  Spring 
Balance. 


;^ 


offers  one  of  the  most  convenient  and  accurate  means  for 
comparing   masses;     thus,   to   make    one    mass   equal    to 

another,    we   have   only  to   adjust  the 
Co)  quantity  of  each  until  they  stretch  the 

spring  of  a  dynamometer,  Fig.  6,  equally; 

or,  as   we    say,    "  weigh   alike "    when 

placed  on  any  weighing  device.     The 

usual  process  of  determining  the  mass 

of    a    body 

consists     in 

placing   it 

upon     one 

pan     of     a 

beam  balance,  Fig.  7,  and 
known  masses,  called 
"weights,"  upon  the  other  pan 
until  the  two  balance.  The 
sum  of  the  known  masses  used 
gives  the  mass  of  the  body.  Such  known  masses,  ranging 
from  one  milligram  up  to  several  hundred  grams,  consti- 
tute a  so-called  "set  of  weights." 

12.  Density.  —  Everyday  experience  teaches  us  that 
bodies  may  have  the  same  size  and  yet  differ  greatly  in 
weight.  A  bar  of  iron,  for  example,  is  much  heavier 
than  a  bar  of  wood  of  the  same  dimensions,  because  its 
mass  is  much  greater.  The  substances  are  said  to  differ 
in  density.  The  density  of  a  substance  is  measured  by  the 
number  of  units  of  mass  contained  in  a  unit  of  volume.- 
Thus  in  the  C.  G.  S.  system  it  is  expressed  as  the  number 
of  grams  per  cubic  centimeter;  in  the  common,  or  foot- 
pound-second (F.  P.  S.),  system,  by  the  number  of  pounds 
per  cubic  foot.  For  example,  the  density  of  lead  is  11.36 
grams  per  cubic  centimeter  (abbreviated  11.36  g./crn.3). 


FIG.  7.  —  A  Beam  Balance. 


INTRODUCTION 


EXERCISES 

1.  Find  the  density  of  a  liquid  of  which  a  liter  has  a  mass  of  850  g. 

2.  If  the  volume  of  a  piece  of  glass  whose  mass  is  10  g.  is  3.9  cu. 
cm.,  what  is  the  density  of  the  glass  ? 

3.  The  density  of  mercury  is  13.6  g./cm.8     Calculate  the  mass  of 
mercury  that  can  be  contained  in  a  vessel  whose  capacity  is  30  cm.8 

4.  If  mercury  is  90  ct.  per  pound,  what  will  half  a  liter  cost? 

5.  A  vessel  will  hold  500  g.  of  mercury.    How  many  grams  of  water 
will  bb  required  to  fill  it  ? 

6.  The  diameter  of  a  steel  sphere  is  4  cm.     If  the  density  of  steel 
is  7.8  g./cm.3,  what  is  the  mass  of  the  sphere?     Volume  of  a  sphere 


13.  States  of  Matter.  —  Matter  admits  of  being  sepa- 
rated into  three  classes  according  to  its  ability  to  preserve 
(1)   its    shape    and    volume,    (2)    its   volume    only,    or 
(3)  neither  its  shape  nor  volume.     A  body  that  retains 
both  its  shape  and  volume  is  called  a  solid;  one  that  re- 
tains its  volume  only  and  shapes  itself  to  the  vessel  con- 
taining it,  a  liquid;  while  one  that  occupies  completely 
any  vessel  in  which  it  is  placed,    retaining  neither  form 
nor  size,  is  called  a  gas.     Ice,  water,  and  steam  are  exam- 
ples of  the  same  substance  in  the  three  states. 

14.  Time.  —  All  systems  of  measurement  of  time,  used 
in  Physics,  employ  the  interval  called  the  mean  solar  second 
as  the  unit.     It  is  -gg-g^  of  a  mean  solar  day,  the  average 
length  of  time  intervening  between  two  successive  transits 
of  the  sun's  center  across  a  meridian. 

SUMMARY 

1.  Physics  is  the  science  of  phenomena  (§  1). 

2.  Matter  is  always  recognized  by  its  properties  of  ex- 
pansion, indestructibility,  and  weight.     Different  kinds  of 
matter  —  gold,  water,  air,  etc.,  are  recognized  by  their 
properties;  i.e.  hardness,  fluidity,  etc.  (§  3). 


12  A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

3.  The  quantity  of  matter  in  a  body  is  called  its  mass. 

4.  There  are  two  well-known  systems  of  measurement 
for  the  quantities  of  length,  area,  volume,  time,  etc.,  viz. 
the  English  system  and  the  metric  system  (§  4). 

5.  The  units  of  the  metric  system  are  the  centimeter, 
gram,  and  second.      This  system  is  the  more  generally 
used    for   scientific   work.      It    is   called   the    centimeter- 
gram-second  (C.  G.  S.)  system  (§  5). 

6.  Units  of  the  English  system  are  the  foot,  pound,  and 
second.     This  system  is  therefore  called  the  foot-pound- 
second  (F.  P.  S.)  system  (§  12). 

7.  The   terms   mass   and   weight  have  not   the   same 
meaning.     The  weight  of  a  body  refers  to  the  downward 
pull  of  the  earth  upon  the  body.     The  mass  of  a  body 
may  remain  constant,  while  the  weight  varies  with  the 
latitude,  the  altitude  above  the  earth's  surface,  and  the 
depth  to  which  it  may  be  lowered  into  the  earth  (§  10). 

8.  Masses  are  measured  and  compared  by  the  process 
of  weighing  (§  11). 

9.  The  density  of  a  substance  is  measured  by  the  num- 
ber of  units  of  mass  contained  in  a  unit  of  volume  (§  12). 

10.  Matter  may  be  divided  into  three  classes  according 
to  its  ability  to  preserve  (1)  its  shape  and  volume,  (2)  its 
volume  only,  or  (3)  neither  its  shape  nor  volume.  Thus 
bodies  are  classed  as  solids,  liquids,  and  gases  (§  13). 


CHAPTER   II 

MOTION,  VELOCITY,   AND   ACCELERATION 
,  l.    UNIFORM  MOTIONS  AND  VELOCITIES 

15.  Motion  and  Rest ~  Relative  Terms,  — The  position  of 
a  body  at  any  instant  is  defined  by  its  direction  and  dis- 
tance from  some  other  body  which  is  usually  conceived  as 
being  fixed,  or  at  rest*.    Motion  is  a  continuous  change  in 
the  position  of  a  body.     It  is  customary  to  think  of  the 
earth  as  the  fixed  body  when  we  speak  of  the  motion  of  a 
train,  a  bird,  a  cloud,  etc.     Again,  a  passenger  sitting  in 
a  moving  railway  coach  is  in  the  condition  of  rest  with 
respect  to  the  train,  while  with  respect  to  the  earth  the 
same  person  is  in  rapid  motion.     Even  the  earth,  as  we 
know,  is  not  at  rest;  it  not  only  rotates  on  its  axis,  but 
travels  with   great   speed   in   its  orbit   around    the  sun. 
Hence  a  body  at  rest  with  respect  to  the  earth  is  not  actu- 
ally at  rest,  nor  is  its   motion  with  respect  to  the  earth 
the  actual  motion.     However,  when  no  statement  is  made 
to  the  contrary,  the  earth  is  regarded  as  the  body  to  which 
the  motion  of  an  object  is  referred* 

16.  Path  of  a  Moving  Body.  —  The  line  described  by  a 
small  moving  body  is  called  its  path.     When  the  path 
described  is  a  straight  line,  the  motion  is  rectilinear ;  when 
curved,  the  motion  is  curvilinear.     Let  us  first  consider 
cases  of  rectilinear  motion, 

17.  Uniform  Rectilinear  Motion, — If  a  moving  body  de- 
scribes equal  portions  of  its  path  in  equal  intervals  of  time, 
no  matter  how  small  the  intervals  may  be,  its  motion  is  uni- 
form.    In  other  words,  uniform  motion  is  the  motion  of  a 

13 


14  A  HIGH   SCHOOL  COURSE  IN  PHYSICS 

body  when  the  distance  passed  over  is  proportional  to  the 
time  occupied.  In  the  case  of  uniform  rectilinear  motion 
the  velocity,  or  rate  of  motion,  of  the  moving  body  is  con- 
stant in  both  magnitude  and  direction,  and  is  measured  by 
the  distance  which  the  body  travels  per  second,  per  minute, 
per  hour,  etc.;  for  example,  10  centimeters  per  second 
(abbreviated  10  cm. /sec.),  25  miles  per  hour,  etc. 

18.  Equation  of  Uniform  Motion.  — From  §  17  it  can  easily 
be  seen  that  the  entire  distance  passed  over  by  a  body  having 
uniform  motion  can  be  found  by  multiplying  the  velocity  by 
the  time.    Thus,  if  the  velocity  is  20  cm. /sec.,  the  distance 
passed  over  in  five  seconds  is  5  x  20,  or  100,  centimeters. 
This  relation  between  the  distance  d,  the  velocity  v,  and 
the  time  t  is  conveniently  expressed  by  the  equation 

d=vt.  (l) 

19.  Representation  of   a   Motion.  —  In  describing  com- 
pletely the  rectilinear   motion  of   a   body,  the  following 
characteristics   must   be   given:     (a)    the   starting  point, 
(5)  the  direction,  and  (e)  the  distance  traveled.     It  will 
be  observed  that  a  straight  line,  since  it  has  origin,  direc- 
tion, and  length,  is  capable  of  representing  the  three  char- 
acteristics  of   rectilinear   motion.     Hence   the   line  AB, 
Fig.  8,  drawn  from  the  point  A  a  distance  of  4  centimeters 

to  the  right  may  be  used   to 
d , §    represent  the  three   qualities 

FIG.  8.— Representation  of  a         of    the    motion    of    a    body    4 

miles  in  an  easterly  direction 

from  the  place  represented  by  the  point  A.  By  letting  a 
centimeter  represent  5  miles  the  same  line  will  represent 
the  characteristics  of  the  rectilinear  motion  of  a  body  over 
a  distance  of  20  miles.  Thus  any  convenient  scale  may 
be  used,  but  the  same  scale  should,  of  course,  be  used 
throughout  a  given  problem. 


MOTION,   VELOCITY,   AND   ACCELERATION          15 

20.  Average  Velocity.  —  Absolute  uniform  motion  is  of 
very  rare  occurrence,  except  during  exceedingly  small  in- 
tervals of  time^    For  instance,  a  train  on  leaving  a  station 
starts  slowly  and  gains  in  speed  until  it  acquires  the  veloc- 
ity with  which  it  can  follow  schedule  time.    •  It  would  be 
practically  impossible  for  the  engineer  so  to  regulate  the 
throttle  as  to  maintain  an  absolutely  constant  velocity,  in- 
asmuch as  the  resistance  due  to  the  track,  curves,  wind, 
etc.,  would  vary  from  time  to  time.  "*  Finally  the  throttle 
is  closed,  the  brakes  applied,  and  the  tijain  comes  gradually 
to  rest.     Although  the  velocity  has  changed  greatly,  the 
train  has  passed  over  a  certain  distance  in  a  definite  time; 
let  us  say  30  miles  in  40  minutes.     The  train  has  moved 
just  as  far  as  it  would  have    traveled  with  a   uniform 
velocity  found  by  dividing  the  total  distance  by  the  time 
consumed,  i.e.  -|  of  a  mile  per  minute.     This  is  called  the 
average  velocity  of  the  train  for  the  time  under  considera- 
tion.    From  this  explanation  it  is  obvious  that  equation 
(i)  will  hold  for  motions  that  are  not  uniform,  provided 
v  is  the  average  velocity  for  the  time  t. 

21.  Velocity  at  Any  Instant.  — If  we  examine  the  motion 
of  all  the  objects  with  which  we  are  familiar,  we  shall  find, 
as  in  the  case  of  the  train  in  §20,   that  the  velocity  in 
almost  every  instance  is  either  increasing  or  decreasing. 
Hence  velocity  cannot  be  defined  accurately  as  the  distance 
over  which,  a  body  moves  in  a  unit  of  time.     We  must, 
therefore,  consider  the  velocity  of  a  body  at  a  given  in- 
stant, i.e.  at  some  stated  time.      The  velocity  of  a  body  at 
a  given  instant  is  the  distance  it  would  move  in  a  second  if  at 
that  instant  its' motion  were  to  become  uniform. 

22.  Representation  of  a  Velocity.  —  A  velocity  has  the 
.characteristics    of   magnitude    (i.e.    speed)   and  direction. 

Therefore,  a  straight  line  of  a  definite  length  may  conven- 
iently b6  used  to  represent  the  velocity  of  a  body  at  a 


16  A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

given  instant.  Let  the  velocity  of  a  body  be  10  miles  per 
hour  north.  A  straight  line,  as  AB,  Fig.  9,  drawn  from 
B  A  to  B  in  an  upward  direction  and  having  a 

length  of  10  units,  will  completely  represent 
•  the  given  velocity.     As  in  the  case  of  mo- 
tions (§19),  any  convenient  length  may  be 
selected  to  represent  a  unit  of  velocity,  but 
the  same  length  should  be  used  throughout 
any    given    discussion.     Any   quantity,    as 
velocity,  having  direction  as  well  as  magni- 
Fia.9.— Repre-   tude,  is  called  a  vector  quantity,  and  the  line 

sentation    of 

a  Velocity,      representing  it,  a  vector. 

EXERCISES 

1.  A  velocity  of  60  mi.  per  hour  is  how  many  feet  per  second  ? 

2.  Express  a  velocity  of  10   m.  per  minute  in  centimeters  per 
second.    Draw  the  vector  that  represents  the  velocity  when  the  direc- 
tion is  eastward. 

3.  A  train  travels  100  mi.  in  two  and  one  half  hours.     Calculate 
the  average  velocity  in  feet  per  second. 

4.  The  speed  of  an  electric  car  averages  2.0  ft.  per  second.     How 
tar  will  it  travel  in  three  hours  ? 

5.  A  train  whose  length  is  440  yd.  has  a  velocity  of  45  mi.  per 
hour.     How  long  will  it  take   the  train  to  pass  completely  over  a 
bridge  100  ft.  long?  Ans.   21.51  sec. 

6.  A  wheel  50  cm.  in  diameter  revolves  600  times  per  minute. 
Express  the  speed  of  a  point  on  its  rim  in  centimeters  per  second. 

Ans.    1570.8  cm. /sec. 

7.  Assuming  that  the  radius  of  the  earth  is  4000  mi.  and  'that  it 
revolves  on  its  axis  once  in  exactly  24  hr.,  ascertain  the  speed  of  a 
point  at  the  equator.  Ans.   1047.2  mi./hr. 

2.    UNIFORMLY  ACCELERATED  MOTION 

23.    Acceleration.  —  We  have  spoken  of  the  manner  in 
which  a  train  leaves  a  station.     Starting  from  rest  and. 
gradually  gaining  in 'speed,  it  finally  attains  the  desired 
velocity.     Let  us  suppose  that  after  the  train  has  been 


MOTION,   VELOCITY,   AND   ACCELERATION          17 

moving  one  second  its  velocity  is  20  cm. /sec.;  at  the  end 
of  two  seconds  from  the  instant  of  starting,  40  cm. /sec.; 
at  the  end  of  three  seconds,  60  cm. /sec.,  etc.  While*  the 
.  train  continues  to  move  in  this  manner,  the  velocity  is 
increasing  the  same  amount  during  each  second,  viz.,  20 
cm. /sec.  This  quantity,  the  rate  at  which  the  velocity 
changes  with  the  time,  is  called  the  acceleration  of  the  train. 
Since  the  change  in  velocity  per  second  is  measured  in 
centimeters  per  second,  the  acceleration  of  the  train  is 
conveniently  expressed  thus:  20  cm. /sec.2,  and  read  "20 
centimeters  per  second  per  second."  The  acceleration  of 
a  body  is  positive  or  negative  according  as  the  velocity 
increases  or  decreases..  The  acceleration  of  a  falling  body, 
for  example,  is  positive ;  that  of  a  body  thrown  upward, 
negative. 

24.  Uniformly  Accelerated  Motion.  — If  the  rate  at  which 
the  velocity  of  a  moving  body  changes  with  the  time  be  con- 
stant, — i,e*  if  the  acceleration  remain  uniform)-*- the  motion 
is  called  uniformly  accelerated  motion.  The  motion  of  the 
train  considered  in  §  23  is  of  this  t}^pe.  There  occur  in 
-  nature  many  cases  in  which  the  condition  that  defines 
uniformly  accelerated  motion  is  very  nearly  fulfilled ;  e.g. 
falling  bodies,  bodies  thrown  upward,  bodies  moving  freely 
along  inclined  planes,  etc. 

0.  Velocity  Acquired  and  Distance  Traversed.  —  The 
velocity  acquired  in  a  given  number  of  seconds  by  a  body 
having  uniformly  accelerated  motion  is  found  in  a  manner 
which  the  following  example  clearly  illustrates : 

A  body  starts  from  rest  and  gains  in  speed  4  cm./see.  in 
each  second  of  time. 

At  the  end  of  one  second  its  velocity  is  4  cm.  per  second. 
At  the  end  of  two  seconds  its  velocity  is  8  cm.  per  second. 
At  the  end  of  three  seconds  its  velocity  is  12  cm.  per  second. 
At  the  end  of  t  seconds  its  velocity  is  4 1  cm.  per  second. 


18  A  HIGH   SCHOOL  COURSE  IN  PHYSICS 

It  is  plain,  therefore,  that  the  velocity  v  at  the  end  of 
any  number  of  seconds  t  is  found  by  multiplying  the 
acceleration  a  by  the  time  ;  or. 

v  =  at.  (2) 

Again,  since  the  initial  velocity  for  a  given  interval  of 
time  is  0,  and  the  final  velocity  is  v,  and  since  the  gain  in 
velocity  is  uniform,  the  mean  velocity  for  the  interval  is 
(0  +  v)-*-  2.  If  the  acceleration  is  4  cm./sec.2  as  before^ 

the  average  velocity  for  the  first  second-is     "t"     cm.  per 
second ; 

0  4-  % 
the  average  velocity  for  the  first  two  seconds  is    — ~ —  cm. 

per  second ; 

the  average  velocity  for -the -first  t  seconds  is  -        -  cm. 

2t 

per  second;  and  if  the  acceleration  is  a  cm./sec.2,  the  av- 
erage velocity  for  t  seconds  is  —~-  cm./sec.,  or  —• 

Now  since  the  distance  passed  over  is  found  by  mul- 
tiplying the  average  velocity  by  the  time  (§  20),  the 
distance  that  the  body  moves  in  t  seconds  when  the  accel- 
eration is  4  cm./sec.2  is  —  x  £,  or  J(4tf2)  cm.;  but  if  the 

acceleration  is  a,  we  have  for  the  distance,  ~  x  £,  or  |  at2,. 

z 

Hence  d  =  Jat».  |    (3) 

EXAMPLE. —  A  car  starts  to  move  down  an  incline  that  is  just  sleep 
enough  to  cause  it  to  gain  in  velocity  50  cm./sec.  during  each  second 
of  time.  Find  the  velocity  and  the  distance  traversed  at  the  end  of 
the  fifth  second,  and  the  distance  that  the  car  moves  during  the  fifth 
second. 

SOLUTION.  —  The  initial  velocity  is  0.  The  final  velocity  is  the 
product  of  the  gain  per  second  and  the  number  of  seconds.  Hence 
v  =  50  x  5,  or  250  cm./sec. 


MOTION,  VELOCITY,   AND   ACCELERATION 


19 


The  total  distance  traversed  is  the  product  of  the  average  velocity 
multiplied  by  the  number  of   seconds.     The  average  velocity  for  5 

seconds    is    SjJT "°  ,   and    the    number    of    seconds    is    5.      Hence 


250 


d  =  —  —  -  x  5,  or  625  cm.      The  same  result  could  be  found  by  sub- 

stituting the  values  of  a  and  t  in  equation  (3).  The  analysis  of  a 
problem,  however,  is  far  more  valuable  than  the  mere  substitution  of 
numbers  in  a  formula. 

The  distance   traversed  during  the  fifth  second  is  evidently  the 
difference  between  the  distance  traversed  in  5  seconds  and  that  in 

0  4-  4 


4  seconds.     Now  in  4  seconds  the  car  moves 


x  4,  or  400  cm. 


Hence  during  the  fifth  second  the  car  moves  625  —  400,  or  225  cm. 

We  thus  observe  that  if  any  two  of  the  four  quantities 
used  in  the  discussion  be  given,  the  others  can  be  calcu- 
lated by  the  help  of  equations  (2)  and  (3). 

The  relation  of  time  and  distance  shown  by  equation  (3) 
may  be  tested  experimentally  as  follows  : 

Let  a  grooved  board  AB,  Fig.  10,  about  15  feet  long  be  supported 
with  one  end  elevated  about  18  inches.  The  groove  can  easily  be 
formed  by  nailing  a  strip  of  wood  about  2  inches  wide  to  the  side  of 


u  - 


FIG.  10.  —Verifying  the  Relation  between  the  Distance  Traversed  and  the  Time 
in  Uniformly  Accelerated  Motion. 

a  wider  piece,  forming  a  cross  section  as  shown  in  X.  Sufficient  sup- 
ports should  be  used  to  keep  the  groove  straight.  Arrange  a  seconds 
pendulum  (§  84)  so  as  to  make  and  break  an  electrical  contact  at  the 
center  of  its  path,  and  connect  a  telegraph  sounder  and  a  cell  of 
battery  in  the  circuit  with  the  pendulum.  The  sounder  should  give 


20  A  HIGH   SCHOOL  COURSE  IN  PHYSICS 

a  loud  click  at  the  end  of  every  second.  Release  a  marble  at  A  pre- 
cisely at  the  instant  the  sounder  clicks,  and  place  a  block  C  at  such 
a  point  on  the  incline  that  the  click  of  the  marble  against  C  coincides 
with  the  click  that  marks  the  end  of  the  third  second.  This  point 
will  have  to  be  found  by  trial,  and  should  be  verified  by  two  or  three 
tests.  The  length  A  C  gives  the  distance  passed  over  by  the  marble 
in  three  seconds.  Let  the  process  be  repeated  for  two  seconds  and 
one  second.  The  distances  found  should  be  proportional  to  the 
squares  of  the  tfmes,  as  shown  by  equation  (3)  ;  i.e.  as  1 : 4 : 9. 

EXERCISES 

1.  Solve  both  equations  (2)  and  (3)  for  the  acceleration  a  and  the 
time  t. 

2.  Combine  equations  (2)  and  (3)  so  as  to  express  the  velocity  in 
terms  of  the  acceleration  a  and  the  distance  d.     Express  also  the  dis- 
tance in  terms  of  velocity  and  acceleration. 

3.  Letting  a  line  1  cm.  long  express  the  acceleration  a,  represent 
the  velocity  at  the  end  of  each  of  the  first  four  seconds.     Represent 
also  the  corresponding  distances. 

SUGGESTION.  —  Equation  (2)  gives  the  length  representing  the 
velocity,  and  equation  (3),  the  distance. 

4.  A  train    leaving   a   station  has   a  constant    acceleration    of 
0.4  m. /sec.2.     What  will  be   its  velocity  at  the  end  of   the   tenth 
second?     At  the  end  of  15  seconds? 

5.  If  the  acceleration  of  an  electric  car  is  uniform  and  2  ft./sec.2, 
in  how  many  seconds  will  it  accumulate  a  velocity  of  25   ft.  per 
second  ? 

6.  How  far  will  the  car  in  Exer.  5  move  during  the  first  10  seconds? 
What  will  be  its  average  velocity  during  this  interval  of  time  ? 

7.  The  acceleration  of  a  car  is  5  m./sec.2.     What  velocity  will  it 
acquire  in  going  100  m.?  x       Ans.  31.62  m./sec.2. 

8.  A  body  has  uniformly  accelerated  motion.     What  is  its  accel- 
eration if  it  passes  over  300  cm.  in  20  seconds?         Ans.  1.5  cm./sec.2. 

9.  A  bicycle  starts  from  rest  at  the  top  of  a  hill  150  ft.  long  and 
has  a  uniform  acceleration  of   1  ft.  per  second.     What  will  be  its 
velocity  at  the  foot  of  the  hill  ?  Ans.  17.32  ft./sec. 

10.  A  car  was  moving  at  the  rate  of  30  mi.  per  hour  when  the 
brakes  were  applied.  What  was  the  rate  of  retardation  if  the  car 
came  to  rest  in  10  seconds,  the  decrease  in  velocity  being  uniform  ? 

Ans.  4.4  ft./sec.2. 


MOTION,  VELOCITY,   AND   ACCELERATION          21 

11.  A  bicycle  rider  moving  at  the  rate  of  15  mi.  per  hour  applies 
the  brake  which   brings  him  to  rest  in  moving  121  ft.     Assuming 
that  the  velocity  decreases  uniformly,  find  the  acceleration. 

Ans.    -  2  ft. /sec.2. 

12.  A  fly  wheel  is  set  in  motion  with  a  uniform  acceleration  of 
two  revolutions  per  second  per  second.     If  the  diameter  of  the  wheel  - 
is  50  cm.,  what  is  the  linear  acceleration  of  a  point  on  its  rim  ? 

Am.   314.16  crn./sec.2. 

13.  What  is  the  velocity  of  a  body  having  uniformly  accelerated 
motion  at  the  beginning  of   the  tth  second?    What  is  the  average 
velocity  during  the  t  th  second  ?     Show  that  the  distance  passed  over 
during  the  t  th  second  is  £  a  (2  t  —  1). 

14.  Apply  the  formula  developed  in  Exer.  13  to  the  conditions 
given  in  Exer.  4,  and  calculate  the  distance  passed  over  by  the  train 
during  the  fifth  and  the  tenth  second.  Ans.   1.8  m.  and  3.8  m. 

3.    SIMULTANEOUS  MOTIONS  AND  VELOCITIES 

26.  Composition  of  Motions.  —  The  actual  displacement 
of  a  body  is  often  due  to  two  or  more  causes  acting  to-  \ 
gether.     For  example,  a  ball  rolled  along  the  deck  of  a 
vessel  has  a  displacement  that  is  the  result,  of  combining 
the  displacement  of  the  boat  with  that  given  the  ball  by  the 
hand.     Hence,  to  find  the  displacement  of  the  ball  with 
respect  to  the  earth,  we  must  take  into  account  all  the 
separate  motions  that  enter  into  the  case.     It  will  be  ob- 
served that  different  cases  will  arise  depending  on  the  rela- 
tion of  the  magnitudes  and  directions  of  the  displacements 
to  be  compounded.^    The  individual  motions  effecting  the 
displacement  are  called  the  components,  and  the  motion 
due  to  the  united  action  of  the  components,  the  resultant. 
The  process  of  finding  the  resultant  from  the  components 
is  called  the  composition  of  motionsv 

27.  Compounding  Motions  in  a  Straight  Line.  —  Let  a 
ball  be  rolled  along  the  deck  of  a  vessel  toward  the  bow. 
While  the  ball  moves  over  a  distance  of  20  feat  along  thfi 

the  boat  mxiYes  forward  30  feet.     Since  the  com- 


22  A  HIGH   SCHOOL  COURSE  IN  PHYSICS 

ponent  motions  are  in  the  same  direction,  it  is  plain  that 
the  ball  actually  moves  a  distance  of  50  feet  in  the  direction 
the  vessel  is  going.  Hence  the  resultant  is  equal  to  the 
sum  of  the  components.  Again,  imagine  that  the  ball  is 
rolled  toward  the  stern,  a  distance  of  20  feet,  while  the 
boat  is  moving  forward  30  feet.  In  this  case  we  can  see 
that  the  ball  will  be  carried  forward  by  the  vessel  10  feet 
farther  than  the  distance  which  it  rolls  backward.  Hence 
the  resultant  is  10  feet,  and  in  the  direction  of  the  motion 
of  the  vessel,  because  that  is  the  larger  component.  A 
rule  for  compounding  motions  in  the  same  straight  line 
may  be  stated  as  follows: 

The  resultant  of  two  component  motions  in  the  same  straight 
line  is  equal  to  their  sum  when  the  directions  are  the  same, 
and  to  their  difference  when  the  directions  are  opposite.  In 
the  latter  case  the  resultant  is  in  the  direction  of  the  greater 
component. 

28.  Compounding  Velocities  in   a  Straight    Line.  —  If 
the  velocities  of  the  boat  and  the  ball  considered  in  §  27 
are  respectively  15  and  10   feet  per  second,  and  if  the 
directions  are  the  same,  it  is  clear  that  the  actual  velocity 
of  the  ball  will  be  the  sum  of  the  velocity  of  the  boat  and 
the  velocity  given  to  the  ball  by  the  hand,  or  25  feet  per 
second;   and   if   the   directions   are    opposite,  the  actual 
velocity  of  the   ball  will  be  equal    to   the   difference  of 
the  velocities,  or  5  feet  per  second.     In  the  latter  case 
the  resultant  velocity  is  in  the  direction  of  the  boat's 
motion,  since  that  is  the  greater  of  the  two  components. 

29.  Compounding  Motions  at  an  Angle.  —  Let   a   man 
starting  from  the  point  A,  Fig.   11,  row  a  boat  perpen- 
dicular at  all  times  to  the  current  of   a  river  40  rods 
in  width.     If  there  were  no  current,  the  boat  would  land 


MOTION,   VELOCITY,   AND   ACCELERATION          23 


at  B.  But,  while  crossing,  the  current  carries  the  boat 
down  the  stream  a  distance  AC,  which  we  may  call  30 
rods.  The  boat  therefore  lands  at  J9,  having  taken 'the 
path  AD.  Since  AD  is  in 
this  case  the  diagonal  of  a 
rectangle  whose  sides  are 
40  and  30  units,  represent- 
ing distances  measured  in 
rods,  its  length,  which  is 
50  units,  represents  a  dis- 
tance of  50  rods,  the  re- 
sultant motion  of  the  boat. 
If  the  angle  between  the 
components  is  not  a  right 
angle,  a  boat  starting  from 
E  takes  a  path  Uff,  the  di- 
agonal of  an  oblique  paral- 
lelogram EFH&. 


FIG.  11.  —  Resultant  of  Two  Motions 
at  an  Angle. 


A  thin  piece  of  wood  E,  Fig. 
12,  is  arranged  to  slide  smoothly 
along  the  edge  of  a  drawing  board.  At  m  and  n  wire  nails  are  driven 

a  short  distance  into  the  wood  and  a 
third  into  the  board  at  o.  Loop  one 
end  of  a  piece  of  thread  around  m, 
pass  it  over  n,  and  attach  the  other 
end  to  a  small  weight  at  A .  If  the 
board  is  now  placed  in  a  vertical  po- 
sition and  the  slide  moved  from  E  to 
E',  the  weight  undergoes  a  displace- 
ment AB.  If  the  loop  is  transferred 
to  nail  o,  on  the  stationary  board,  a 
movement  of  the  slide  from  E  to  E' 
gives  the  weight  two  simultaneous 
displacements  represented  by  AB  and  BD,  causing  it  to  follow  the 
diagonal  path  AD.  If  the  operations  are  repeated  after  tilting  the 
board  in  its  plane,  it  will  be  seen  that  the  weight  follows  the  diagonal 


FIG.  12. —  Apparatus  for  Show- 
ing the  Compounding  of  Two 
Motions. 


24  A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

of  an  oblique  parallelogram  as  the  result  of  the  two  component  dis- 
placements. 

The  facts  shown  by  this  experiment  may  be  stated  as  follows  : 

The  resultant  of  two  component  uniform  motions  not  in  the 
same  straight  line  is  represented  by  the  diagonal  of  a  paral- 
lelogram whose  adjacent  sides  represent  the  two  component- 
motions. 

30.  Compounding  Velocities  at  an  Angle.  — Velocities 
may  be  compounded  in  the  same  manner  as  motions.     For 
example,  if  the  velocity  with  which  an  oarsman  rows  his 
boat  is  represented  in  magnitude  and  direction  by  the  line 
EF  in  Fig.  11,  and  the  velocity  of  the  stream  by  the  line 
EGr,  the  actual  velocity  of  the  boat  is  represented  by  the 
line  EH. 

It  should  be  observed  that  in  cases  where  the  angle  between 
the  two  components  is  a  right  angle,  the  resultant  is  the  square  root 
of  the  sum  of  the  squares  of  the  components.  In  other  cases  the 
parallelogram  should  be  constructed  accurately  and  the  diagonal  care- 
fully measured. 

31.  Compounding  Several  Motions.  --  When  the  actual 
motion  of  a  body  is  due  to  the  united  action  of  more  than 
two  components,  the  final  resultant  is  found  by  first  deter- 
mining   the    resultant 
of  any  two  of   them; 
and   this    resultant   is 
then  compounded  with 
a  third  component,  and 
so  on  until  each  com- 
ponent has  been  used. 

Let  AB,  AC,  and  A D, 
Fig.  13,  be  three  component 
motions    imparted     simul- 
of   taneously  to  a  body  at  A. 
Three  Component  Motions.  The  resultant  of  any  two, 


MOTION,   VELOCITY,   AND   ACCELERATION          25 

e.g.  AR  and  AD,  is  found  in  the  manner  described  in  §29,  giving  the 
resultant  AE.  AE  is  now  treated  as  a  component  and  compounded 
with  the  third  component  A  C.  In  this  construction  A  CFE  is  the  par- 
allelogram of  which  AF  is  the  diagonal.  AF  is  the  resultant  of  the 
three  given  components. 

32.  Resolution  of  Motions  and  Velocities.  —  The  meaning 
of  this  process,  which  is  the  reverse  of  composition,  is  most 
readily  understood  after  considering  a  particular  case. 
For  example,  let  it  be  required  to  find  the  easterly  velocity 
of  a  vessel  sailing  with  a  velocity  of  15  miles  per  hour  in 
a  direction  east  by  30°  south.  component 


Let  AB,  Fig.   14,   represent   the 
given  velocity  of  the  vessel,  making 
the  angle  BA  C  equal  to  30°.     If  the 
lines  BD  and  BC  are  now  drawn 
parallel  to  A  C  and  AD  respectively,      D]'- 
the   line  AC  represents    the    com-        ' 
ponent  velocity  of  the  vessel  in  an     FIG.  14. -Resolution  of  a  Velocity. 
easterly  direction.     AD  is  the  southerly  component. 

EXERCISES 

1.  A  train  approaches  Chicago  with  a  velocity  of  30  km.  per  hour, 
while  a  brakeraan  runs  along  the  tops  of  the  cars  toward  the  rear  at 
the  rate  of  5  km.  per  hour.     How  ra"pidly  is  the  brakeman  approach- 
ing Chicago? 

2.  A  boy  is  paddling  a  canoe  along  a  river  in  the  direction  of  the 
current,  which  has  a  velocity  of  4  mi.  per  hour ;  if  there  were  no  cur- 
rent, the  canoe  would  move  3.5  mi.  per  hour.     How  fast  is  the  canoe 
moving?  . 

3.  Suppose  the  boy  in  Exer.  2  should  double  his  effort  and  paddle 
upstream.     How  long  would  it  take  him  to  go  10  mi.  ? 

4.  A  ship  is  moving  east  at  the  rate  of  15  mi.  per  hour.     If  a 
person  walks  directly  across  the  deck  at  the  rate  of  4  mi.  per  hour, 
with  what  velocity  will  he  actually  move  ? 

5.  A  boat  is  rowed  with  a  velocity  of  4  mi.  per  hour,  perpendicu- 
lar to  the  current  of  a  stream  flowing  5  mi.  per  hour.     Determine  the 
direction  of  the  motion  and  the  velocity  of  the  boat. 

6.  A  ship  headed  due  east  under  a  power  that  can  move  it  12  mi. 


26  A  HIGH   SCHOOL  COURSE  IN  PHYSICS 

per  hour  enters  an  ocean  current  whose  velocity  is  4  mi.  per  hour  south. 
If  a  person  on  deck  walks  northeast  with  a  velocity  of  3  mi.  per  hour, 
what  is  his  actual  velocity?  Ans.  14.3  mi./hr. 

7.  A   balloon  is  driven   in  a  direction  east  by  30°  north.     How 
rapidly  is  it  drifting  north  if  its  velocity  is  20  mi.  per  hour  ? 

Ans.  10  mi./hr. 

8.  A  body  moves  down  an  inclined  plane  5  m.  in  length.     If  the 
angle  between  the  incline  and  a  horizontal  plane  is  60°,  what  are  the 
horizontal  and  vertical  components  of  its  motion? 

9.  How  rapidly  is  a  bird  approaching  the  equator  when  flying 
due  southeast  at  the  rate  of  20  mi.  per  hour  ? 

SUMMARY 

1.  Motion  is  a  continuous  change  in  the  position  of  a 
body.     Motion  and  rest  are  relative  terms.     When  the 
motion  of  terrestrial  bodies  is  under  consideration,  the 
earth  is  usually  regarded  as  being  at  rest  (§  15). 

2.  The   motion   of   a  body  is  rectilinear  or   curvilinear 
according  as  the  path  described  by  the  body  is  a  straight 
or  a  curved  line  (§  16). 

3.  When  a  body  moves   over   equal  spaces   in   equal 
periods  of  time,  no  matter  how  small  the  period  may  be, 
the  motion  is  said  to  be  uniform.     When  the  motion  of  a 
body  is  uniform,  the  distance  passed  over  is  proportional 
to  the  time  (§  17). 

4.  The  equation  of  uniform  motion  is  d  =  vt  (§  18). 

5.  The    characteristics  of  the  rectilinear  motion  of  a 
body  are  its  starting  point,  the  direction  of  the  motion,  and 
its  displacement.    These  three  qualities  may  be  represented 
by  a  straight  line  (§  19). 

6.  The  average  velocity  of  a  body  is  found  by  dividing 
the  space  passed  over  by  the  time  consumed  (§  20). 

7.  The  velocity  of  a  body  at  any  instant  is  measured 
by  the  distance  it  would  nuwe  in  a  second  if  at  that  instant 
its  motion  were  to  become  uniform  (§  21). 


MOTION,  VELOCITY,   AND  ACCELERATION          27 

*• 

8.  The  characteristics  of  a  velocity  are  its  magnitude 
(or  speed)    and   direction.     The  qualities  may  be  repre- 
sented by  the  length  and  direction  of  a  straight  line.     A 
line  used  in  this  manner  is  called  a  vector  (§  22). 

9.  The  rate  at  which  velocity  changes  with  the  time 
is  called  acceleration  (§  23). 

10.  A  body  has  uniformly  accelerated  motion  when  its 
velocity  changes  at  a  uniform  rate.    The  velocity  acquired 
in  a  given  time  by  a  body  starting  from  rest  may  be  found 
from  the  equation  v  —  at.     The  distance  passed  over  is 
given  by  the  equation  d  =  ^at2  (§  24). 

11.  The  simultaneous  individual  motions  (or  velocities) 
of  a  body  are  called  the  components  of  its  motion  (or  ve- 
locity), and  the  motion  (or  velocity)  due  to  the  united 
action  of  the  components  is  called  the  resultant.     The  pro- 
cess of  finding  the  resultant  from  the  components  is  called 
the  composition  of  motions  (or  velocities)  (§§  26  and  28). 

12.  The  resultant  of  two  simultaneous  motions  (or  veloci- 
ties) along  the  same   straight   line  is  their  sum  when  the 
directions  are  the  same,  and  their  difference  when  they 
are  opposite  (§§  27  and  28). 

13.  The  resultant  of  two  simultaneous  motions  (or  veloci- 
ities)  not  in  the  same  straight  line  is  represented  in  magni- 
tude and  direction  by  the  diagonal  of  a  parallelogram 
whose  adjacent  sides  represent  the  two  components  (§§  29 
and  30). 

14.  The  resultant  of  more  than  two  simultaneous  motions 
(or  velocities)  is  found  by  compounding  the  resultant  of 
any  two  of  them  with  a  third  component,  then  this  new 
one  with  the  fourth,  and  so  on  until  each  component  has 
been  used  (§  31). 

15.  The  process  of  finding  the  components  from  the 
resultant  is  called  resolution  (§  32). 


CHAPTER   III 

LAWS  OF  MOTION  — FORCE 

1.    DISCUSSION  OF  NEWTON'S  LAWS 

33.  Momentum.  —  It  is  a  well-known  fact  that  a  mov- 
ing body  must  always  have  been  put  in  motion  by  an  effort 
on  the  part  of  some  agent.  The  amount  of  this  effort  de- 
pends (1)  on  the  mass  of  the  body  moved  and  (2)  on  the 
rapidity  with  which  it  is  given  velocity,  i.e.  on  the  acceler- 
ation. 3  If  we  observe  a  locomotive  as  it  starts  a  train,  we 
readily  see  that  the  effort  required  is  greater  as  the  train 
is  longer  or  more  heavily  loaded.  Furthermore,  in  order 
to  start  the  train  more  quickly,  a  greater  effort  is  requked 
and  a  more  powerful  engine.  •  Again,  after  the  train  has 
acquired  its  running  speed,  an  effort  is  required  if  the 
motion  is  to  be  destroyed  and  the  train  brought  to  rest. 
This,  also,  depends  on  the  mass  of  the  train  and  the  rapid-v 
ity  with  which  its  motion  is  reduced. 

The  two  quantities,  mass  and  velocity,  determine  what  is 
called  the  quantity  of  motion  in  a  body,  or  its  momentum. 
Momentum  is  measured  by  the  product  of  the  mass  and  the 
velocity  of  a  body,  and  is  expressed  algebraically  as  mv. 
For  example,  the  momentum  of  a  10-gram  rifle  ball  mov- 
ing with  a  velocity  of  25,000  cm. /sec.  is  10  x  25,000,  or 
250,000  C.  G.  S.  units.  No  name  is  used  for  the  unit  of 
momentum. 

•"  34.  Newton's  First  Law  of  Motion.  —  An  inanimate  body 
never  puts  itself  in  motion.  Not  only  does  a  body  with- 
out motion  tend  to  remain  in  that  condition,  but  on  ac- 


LAWS   OF   MOTION  — FORCE  29 

count  of  that  same  tendency  resists  the  effort  of  any  agent 
that  tries  to  start  it.  ~J  On  the  other  hand,  a  body  in  mo- 
tion manifests  a  tendency  to  keep  moving  and  resists  any 
effort  made  to  change  its  motion  in  any  way.  These 
facts  may  be  illustrated  by  the  following  experiments: 

1.  Stand  a  book  upon  end  on  a  sheet  of  paper  placed  flat  upon  the 
table.     Grasp  the  paper  and  try  by  a  quick  pull  to  give  the  book  a 
forward  motion.     On  account  of  the  tendency  of  the  book  to  remain 
at  rest,  it  will  be  found  to  fall  backward.     Repeat  the  experiment, 
but  move  the  paper  very  slowly  at  first ;  and,  while  the  book  is  in 
motion,  let  the  paper  suddenly  stop.     On  account  of  the  fact  that  the 
book  tends  to  remain  in  motion,  it  will  fall  forward. 

2.  Place  a  card  upon  the  tip  of  a  finger  and  lay  a  small  coin  upon 
the  card  directly  above  the  finger  tip.     With   the  other  hand  give 
the  card  a  sudden  snap  in  such  a  manner  as  to  drive  the  card  from 
beneath  the  coin.     The  coin  will  be  left  upon  the  finger.     The  same 
experiment  may  be  varied  by  placing  a  card  upon  the  top  of  a  bottle 
and  a  marble  upon  the  card.     The  sudden  removal  of  the  card  leaves 
the  marble  resting  in  the  mouth  of  the  bottle. 

The  first  to  express  these  facts  of  common  observation 
in  the  language  of  Physics  was  Sir  Isaac  Newton1  (1642- 
1727),  professor  of  mathematics  at  Cambridge,  England. 
The  statement  of  his  First  Law  of  Motion  is  as  follows: 

Every  body  of  matter  continues  in  its  state  of  rest  or  of 
uniform  motion  in  a  straight  line,  except  in  so  far  as  it  is 
compelled  by  force  to  change  that  state? 
t  This  is  known  as  the  Law  of  Inertia,  the  tendency  of 
matter  to  act  in  the  manner  stated  being  often  ascribed  to 
a  property  of  matter  called  inertia.  In  this  law  Newton 
has  given  a  definition  of  force  as  that  which  is  able  to 
cause  or  change  motion.  Hence  the  term  force  is  the  name 

1  See  portrait  facing  p.  30. 

2  "  Every  body  perseveres  in  its  state  of  rest,  or  of  uniform  motion  in 
a  right  line,  unless  it  is  compelled  to  change  that  state  by  forces  impressed 
thereon."  —  Newton's  Principia,  Motte's  Translation. 


30  A   HIGH   SCHOOL  COURSE   IN   PHYSICS 

given  to  the  cause  that  produces  acceleration,  retardation,  or 
a  change  in  the  direction  of  the  motion  of  a  body. 

35.  Newton's   Second  Law   of   Motion.  —  According   to 
Newton's  First  Law  a  moving  body   which  could  be  en- 
tirely freed  from  the  action  of  all  forces  would  have  uni- 
form motion.     A  stone  thrown  from  the  hand  would  take  a 
perfectly  straight  course,  and  a  bullet  fired  upward  would 
never  return  to  the  earth.     The  curved  path  described  by 
the  stone,  however,  indicates  that  a  force  is  acting  upon 
the  body  while  it  is  moving.     This  force,  as  we  know,  is 
the  force  of  gravity. 

Just  as  the  First  Law  defines  force,  so  the  Second  Law 
leads  to  the  measurement  of  force.  This  law  may  be  stated 
as  follows : 

Change  of  motion,  or  momentum,  is  proportion&Ltp  the 
acting  force  and  takes  place  in  the  direction  in  which  the 
force  acts.1 

The  proportion  always  existing  between  force  and  the 
rate  at  which  it  changes  the  momentum  of  a  body  has  led 
to  the  adoption  of  a  convenient  C.  G.  S.  unit  of  force 
called  the  dyne  (pronounced  dine).  The  dyne  is  that 
force  which,  acting  uniformly  for  one  second,  imparts  one 
C.  Gr.  S.  unit  of  momentum  ;  or,  a  dyne  would  give  a  mass 
of  one  gram  an  acceleration  of  one  centimeter  per  second 
per  second.  This  definition  implies  that  the  body  upon 
which  the  force  acts  is  not  at  all  hindered  in  its  motion 
by  external  resistances,  as  friction,  etc.;  i.e.  the  force  has 
simply  to  overcome  the  inertia  of  the  body. 

36.  Equations  of  Force.  —  The  relation  expressed  in  the 
preceding  section  between  force,  momentum,  and  time  ad- 

1  "  The  alteration  of  motion  is  ever  proportional  to  the  motive  force 
impressed ;  and  is  made  in  the  direction  of  the  right  line  in  which  the 
force  is  impressed."  —Newton's  Principia,  Motte's  Translation. 


SIR    ISAAC    NEWTON    (1642-1727) 

The  name  of  Newton  will  always  be  associated  with  the  subject 
of  gravitation  on  account  of  the  fullness  with  which  he  applies  and 
discusses  his  famous  Principle  of  Universal  Gravitation  in  his  book 
entitled  Principia,  published  in  1687.  The  Principia,  which  ranks  as 
a  mathematical  classic,  treats  of  the  laws  governing  the  motion  of 
bodies  under  various  conditions,  and  especially  of  the  motion  of  the 
planets.  This  work  follows  close  upon  the  achievements  of  Galileo 
and  Kepler  in  astronomical  discovery.  Kepler  had  found  by  obser- 
vation that  the  planets  move  around  the  sun  in  elliptical  paths,  which 
Newton  showed  would  be  the  case  if  between  the  sun  and  each 
planet  there  exists  a  force  which  decreases  as  the  square  of  the 
distance  increases. 

The  Principia  laid  a  firm  and  deep  foundation  for  subsequent 
discoveries  in  the  field  of  astronomy;  it  propounded  and  showed  the 
application  of  a  new  method  of  mathematical  investigation,  the  Cal- 
culus, by  which  alone  it  would  retain  its  position  at  the  head  of 
mathematical  treatises. 

Newton  must  also  be  accredited  with  the  announcement  and  elu- 
cidation of  the  three  laws  of  motion  which  bear  his  name  and  with 
numerous  discoveries  in  Light.  His  book  entitled  Optics  contains 
his  discoveries  and  theories  in  this  subject. 

Newton  was  born  in  Lincolnshire,  England,  in  1642,  graduated 
from  Trinity  College,  Cambridge,  in  1665,  and  at  once  began  to  make 
the  discoveries  in  mathematics  and  physics  which  have  immortal- 
ized his  name.  He  was  professor  of  mathematics  at  Cambridge, 
member  of  Parliament,  and  Master  of  the  Mint.  He  was  knighted 
in  1705,  and  at  his  death,  in  1727,  was  buried  in  Westminster  Abbey. 


LAWS  OF  MOTION  —  FORCE  31 

mits  of  being  expressed  in  a  concise  algebraic  form.  Let 
a  mass  of  m  grams  be  acted  upon  by  a  constant  force  ©f  / 
dynes  for  t  seconds.  If  the  velocity  imparted  to  the  mass 
by  that  force  is  v  cm./sec.,  the  momentum  produced  is  mv 
(§  33).  The  momentum  imparted  per  second  is  found  by 
dividing  the  quantity  mv  by  the  time  t.  We  have  there- 
fore, by  the  definition  of  the  dyne, 


f  (in  dynes)  =  **  (in  ^rams)  x  v 


t  (in  seconds) 
From  Eq.  (2)  §  25,  we  have  v  =  at;  <\  ~  "1 

whence  -  is  the  acceleration  a  produced  by  the  force  /. 

Therefore,  by  substituting  in  (l), 

f  (in  dynes)  =  m  (in  grams)  x  a  (in  cm./sec.2).     (2) 

In  the  English  F.P.S.  system  the  unit  of  force  is  the  poundal.  The 
poundal  is  that  force  which,  acting  uniformly  for  one  second,  imparts  one 
F.  P.  S.  unit  of  momentum.  Hence,  if  the  mass  of  the  body  upon  which 
the  force  acts  is  given  in  pounds,  the  velocity  in  feet  per  second,  and 
the  time  in  seconds,  substitution  in  equation  (1)  will  give  us  the  force 
in  poundals.  A  force  of  one  poundal  is  equivalent  to  13,825  dynes. 

EXAMPLE.  —  What  constant  force  acting  four  seconds  will  give  a 
body  of  15  g.  a  velocity  of  20  cm./sec.  ? 

SOLUTION.  —  The  total  momentum  produced  by  the  force  in  four 
seconds  is  15  x  20,  or  300  C.  G.  S.  units.  The  momentum  imparted 
to  the  body  in  one  second  is  300  -^  4,  or  75  units.  Hence  the  force 
is  75  dynes. 

Another  method  is  to  substitute  the  given  quantities  directly  in 
equation  (1),  first  being  assured  that  all  are  given  in  C.  G.  S.  units. 
The  same  may  be  said  of  the  F.  P.  S.  system  in  obtaining  the  force  in 
poundals  when  the  problems  deal  with  English  units. 

37.  Gravitational  Units  of  Force.  —  A  gram-weight  (or 
a  gram-force)  is  the  downward  pull  exerted  by  the  earth 
upon  a  one-gram  mass.  (See  §10.)  If  the  mass  is  free 
to  fall,  the  acceleration  due  to  gravity  will  be  980 
cm./sec.2,  or  980  C.  G.  S.  units  of  momentum  per  second. 


32  A  HIGH   SCHOOL  COURSE  IN  PHYSICS 

Hence,  by  Eq.  (2),  a  gram-weight  is  equivalent  to  980  dynes. 
Likewise,  a  pound-weight  (or  a  pound-force)  is  the  attrac- 
tion of  the  earth  upon  a  one-pound  mass.  The  accelera- 
tion produced  by  this  force  when  the  body  falls  freely  is 
32.16  ft./sec.2,  and  the  momentum  is  32.16  F.  P.  S.  units 
per  second.  Therefore,  one  pound-weight  is  equivalent  to 
32.16  poundals. 

Since  the  earth's  attraction  varies  with  the  locality,  as 
explained  in  §§10  and  69,  the  gram-weight  and  the  pound- 
weight  change  accordingly.  Hence  these  units  are  called 
gravitational  units  of  force.  On  the  other  hand,  the  dyne 
and  the  poundal,  being  independent  of  gravity,  are  called 
absolute  units. 

Commonly,  use  is  made  of  the  pound  and  the  gram  (i.e. 
pound-weight  and  gram-weight)  as  units  of  force.  Scien- 
tific work  has,  however,  demanded  a  more  unvarying  unit 
and  has  brought  the  dyne  into  extensive  use  in  all  cases 
requiring  accuracy  of  expression.  The  poundal  is  little 
used. 

38.  An  Application  of  the  Second  Law.  —  Newton's 
Second  Law  implies  that  any  force  acting  upon  a  body 
produces  its  own  effect,  whether  acting  alone  or  conjointly 
with  other  forces.  An  interesting  illustration  of  this  may 

be  observed  in  the  fol- 
lowing experiment : 

Cut  notches  A  and  B, 
Fig.  15,  in  two  of  the  cor- 
ners of  a  piece  of  wood  about 
2x8  inches.  By  means  of 
a  large  screw  attach  the  cen- 
ter of  the  block  loosely  to  the 
edge  of  a  table  as  shown,  and 
place  a  marble  in  each  notch. 

FIG.  15. -Illustration  of  Newton's  Second        If  the  end  of  the  block  °P- 
Law  of  Motion.  posite  A    is  struck  with   a 


LAWS  OF  MOTION  —  FORCE 


33 


mallet,  ball  A  will  be  dropped  vertically,  while  ball  B  is  projected  hor- 
izontally. B  is  subject  to  two  forces,  an  impulse  which  projects  it  in 
a  horizontal  direction  and  the  constant  force  of  gravity  acting  along 
a  vertical  line.  The  two  marbles  will  be  found  to  strike  the  floor  at 
the  same  time. 

The  experiment  ihows  that  the  effect  of  gravity  in 
bringing  the  balls  to  the  floor  is  independent  of  the  hori- 
zontal component  of  the  motion;  i.e.  a  given  force  (gravity) 
produces  as  much  "  change  in  momentum  "  in  the  vertical 
direction  in  one  ball  as  in  the  other.  After  ball  B  leaves 
the  block,  its  momentum  in  the  horizontal  direction  suffers 
no  change. 

39.  Newton's  Third  Law  of  Motion.  —  To  every  action 
there  is  always  an  equal  and  opposite  reaction;  or,  the  mu- 
tual actions  of  any  two  bodies  are  always  equal  in  magnitude 
and  oppositely  directed. 

This  law  may  be  illustrated  by  experiment  as  follows : 

Let  two  elastic  wooden  balls  A  and 
B,  Fig.  16,  be  suspended  by  threads  in 
such  a  manner  that  they  just  touch  each 
other  when  stationary.  Draw  A  aside 
and  let  it  fall  against  B.  A  will  be 
brought  to  rest  by  the  impact,  and  B  will 
be  moved  to  the  position  B'. 

In  this  experiment  two  results 
are  apparent :  first,  ball  B  is  acted 
upon  by  a  force  sufficient  to  carry 
it  to  the  position  J9',  arid,  second,  ball  A  loses  an  equal 
amount  of  momentum  in  that  it  is  brought  to  rest.  In 
the  impact  occurs  a  mutual  effect  —  a  force  exerted  by 
A  toward  the  right  upon  J5,  and  an  equal  and  oppositely 
directed  force  from  B  upon  A.  This  process  goes  on  in 
every  case  in  which  force  enters.  A  pressure  of  the  hand 
against  the  table  is  opposed  by  an  equal  pressure  of  the  table 
against  the  hand.  When  a  person  leaps  forward  from  a 
4 


_ 

FIG.  16.  — Action  and  Reac- 

tion  are  Equal  and  Oppo- 


34  A  HIGH   SCHOOL  COURSE   IN  PHYSICS 

boat,  the  boat  is  pushed  in  the  opposite  direction.  When 
a  gun  is  fired,  the  mutual  effect  is  to  give  the  bullet  and 
the  gun  equal  momenta;  or,  in  other  words,  the  mass  of 
the  bullet  multiplied  by  its  velocity  equals  the  mass  of 
the  gun  multiplied  by  its  velocity.  The  velocity  of  the 
gun's  recoil,  or  "kick,"  is  small  because  the  mass  of  the 
gun  is  many  times  greater  than  that  of  the  bullet. 

The  question  often  arises :  "  Does  the  earth  rise  to  meet 
the  falling  apple  ? "  In  the  light  of  the  Third  Law  of 
Motion,  we  must  admit  that  the  action  of  the  earth  which 
draws  the  apple  down  is  accompanied  by  a  reaction  which 
is  operative  for  the  same  length  of  time  upon  the  earth  in 
an  upward  direction.  Hence  the  momentum  given  the 
apple  equals  that  given  the  earth.  On  account  of  the 
enormous  mass  of  the  earth,  however,  the  distance  through 
which  it  rises  to  meet  the  apple  is  infinitesimally  small. 

EXERCISES 

1.  Why  does  a  person  standing  in  a  car  tend  to  fall  backward 
when  the  car  starts,  and  forward  when  it  stops  ? 

2.  Why  does  a  bullet  continue  to  move  after  leav- 
ing a  rifle? 

3.  A  weight  W,  Fig.  17,  is  attached  by  a  cord  B 
to  some  fixed  object.     A  quick  downward  pull  on  a 
similar  cord  A  will  break  the  cord  below  W,  but  a 
steady  pull  will  break  cord  B.     Explain. 

4.  A  blast  of  fine  sand  driven  against  glass  soon 
cuts  away  its  smooth  surface.     Why  ? 

5.  If  a  rifle  ball  is  thrown  against  a  board  placed 
on  edge,  it  will  knock  it  down ;  but  when  fired  from 
a  gun,  it  will  pass  through  the  board  and   leave   it 
standing.     Why  ? 

6.  Explain  why  the  head  of  a  hammer  or  mallet 


FIG  17  can  ^e  driven  on  by  simply  striking  the  end  of  the 

handle. 

7.   Why  can  an  athlete  make  a  longer  "  running  jump  "  than  a 
"  standing"  one? 


LAWS  OF  MOTION  — FORCE  35 

8.  Will  a  stone  dropped  from  a  moving  train  fall  in  a  straight 
line? 

9.  Why  do  moving  railway  coaches  "  telescope  "  in  a  collision? 

10.  Explain  how  heavy  fly  wheels  serve  to  steady  the  motion  of 
machinery,  as  in  the  case  of  the  sewing  machine. 

11.  A  4-gram  rifle  ball  leaves  a  gun  with  a  speed  of  20,000  cm.  per 
second.     Compute  its  momentum. 

12.  Which  has  the  larger  momentum,  a  man  weighing  150  lb., 
walking  10  ft.  per  second,  or  a  boy  weighing  60  lb.  and  running  25  ft. 
per  second  ? 

13.  Which  has  the  greater  momentum,  a  man  weighing  160  lb. 
in  a  railway  coach  moving  30  mi.  per  hour,  or  a  2-ton  stone  moving 
3  ft.  per  second?     Express  the  difference  in  F.  P.  S.  units. 

14.  What  force  acting  for  10  seconds  upon  a  mass  of  200  g.  will 
produce  a  velocity  of  5  cm./sec.?     Express  the  change  of  momentum 
per  second  in  C.  G.  S.  units. 

15.  A  body  whose   mass   is  20  g.   is   given  an   acceleration  of 
45  cm./sec.2.     What  is  the  required  force? 

16.  What  acceleration  will  be   given   to  a  •  mass  of  25  g.  by  a 
constant  force  of  500  dynes  ?     Over  what  distance  will  the  body  move 
in  5  seconds  if  the  force  continues  to  act  ? 

17.  If  the  force  given  in  Exer.  16  ceases  to  act  at  the  end  of  the  5th 
second,  how  far  will  the  body  move  during  the  next  5  seconds  ? 

SUGGESTION.  —  Find  the  velocity  imparted  in  the  first  5  seconds 
and  apply  the  First  Law  of  Motion. 

18.  An  inelastic  ball  of  clay  whose  mass  is  200  g.  has  a  velocity 
of  25  cm./sec.  when  it  collides  with  a  similar  ball  at  rest  whose  mass 
is  50  g.     Find  the  velocity  after  collision. 

SUGGESTION.  —  After  impact  the  two  masses  move  on  as  one  mass 
with  the  momentum  of  the  first  before  collision. 

19.  A  projectile  weighing  100  lb.  is  fired  with  a  velocity  of  1200  ft. 
per  second  from  a  gun  weighing  8  T.     Find  the  velocity  with  which 
the  gun  starts  to  move  backward. 

2.    CONCURRENT   FORCES 

40.  Representation  of  Forces.  —  The  three  characteristics 
of  a  force,  its  point  of  application,  direction,  and  magni- 
tude, can,  as  we  have  seen,  be  represented  by  a  straight 
line.  One  end  of  the  line  shows  the  point  of  application. 


36  A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

the  length  of  the  line  shows  the  magnitude  of  the  force, 
and  the  direction  in  which  the  line  is  drawn  shows  the 
direction  in  which  the  force  acts.  For  example,  (1), 

Fig.  18,  represents 
a  force  of  10  dynes 
acting  northeast 
from  the  point  A. 
The  unit  of  force 

15  Dynes  ,  .      -. 

A' >B      may  be  represented 

(0  by  any  convenient 

FIG.  18.  —  The  Representation  of  Forces  by  Means      lenp-th        but       the 
of  Straight  Lines. 

same  scale  should, 
of  course,  be  used  throughout  a  given  problem. 

In  a  similar  manner,  (2),  Fig.  18,  shows  that  two 
concurring  forces,  AB  and  A  C,  representing  respectively 
15  dynes  east  and  8  dynes  north,  act  upon  the  point  A. 
The  scale  adopted  in  this  case  is  2  millimeters  to  the 
dyne. 

41.  Composition   of   Forces.  —  When   a   body   is   acted 
upon  by  two  forces  at  the  same  time,  it  is  easy  to  imagine 
a  single  force   that  might  be  substituted   for  them  and 
would  ha\&e  the   same  effect.      This  single  force  is  the 
resultant  of  the  two  forces,  which  are  the  components. 
The  process  of  finding  the  resultant  of  two  or  more  component 

forces  is  catted  the  composition  of  forces.  Forces  are  com- 
pounded in  the  same  manner  as  motions  and  velocities 
(§§27-31). 

42.  Forces  Acting   in  a  Straight  Line.  —  When   two 
forces  act  upon  a  body  in  the  same  line  and  in  the  same 
direction,  it  is  clear  that  the  resultant  is  the  sum  of  the 
two  components.     For  example,  if  a  weight  is  to  be  lifted 
by  two  men  pulling  upward  on  a  rope  attached  to  it,  and 
if  one  man  pulls  with  a  force  of  50  pounds  while  the  other 
pulls  with  a  force  of  75  pounds,  the  two  forces  result  in  a 


LAWS   OF  MOTION  — FORCE  37 

single  pull  of  125  pounds.  Hence  the  resultant  is  125 
pounds  and  is  directed  upward. 

When  the  two  forces  act  in  opposite  directions  along 
the  same  line,  the  resultant  is  the  difference  of  the  two 
components.  Thus,  if  one  man  pulls  upward  on  a  weight 
with  a  force  of  75  pounds  while  the  other  pulls  downward 
with  a  force  of  50  pounds,  the  lifting  effect  is  the  same  as 
a  single  force  of  75  —  50,  or  25  pounds.  The  action  of  the 
resultant  is  plainly  in  the  direction  of  the  greater  of  the 
two  components.  A  special  case  of  opposite  forces  is  that 
in  which  the  sum  of  the  components  acting  in  one  direction 
is  equal  to  the  sum  of  those  acting  along  the  same  straight 
line  in  the  opposite  direction.  In  this  case  no  motion  can 
result  from  the  joint  action  of  all  the  forces.  In  other 
words,  the  resultant  is  zero.  The  body  upon  which  such 
forces  act  is  said  to  be  in  equilibrium. 

43.  Forces  Acting  at  an  Angle.  —  If  AB  and  AC,  (2), 
Fig.  18,  represent  forces  of  unequal  magnitudes,  it  is  clear 
that  the  resultant  will  divide  the  angle  between  them,  but 
will  lie  nearer  the  greater  force.  The  actual  magnitude 
and  direction  of  the  resultant  are  found  in  the  same  man- 
ner as  in  the  case  of  the  resultant  of  two  motions  (§  29). 

The  resultant  of  two  concurring  forces  acting  at  an  angle  is 
represented  by  the  diagonal  of  a  parallelogram  constructed  on 
the  two  lines  representing  the  component  forces. 

This  is  one  of  the  most  important  laws  of  mechanics  and 
is  universally  known  as  the  Principle  of  the  Parallelogram 
of  Forces.  The  following  experiment  will  illustrate  the 
truth  of  this  principle: 

Arrange  two  dynamometers  (see  Fig.  6),  before  the  blackboard, 
as  shown  in  Fig.  19.  Let  the  weight  W  be  great  enough  to  produce  a 
large  but  measurable  tension  in  each  of  the  oblique  cords.  Place  a 
rectangular  block  of  wood  against  each  of  the  cords  and  trace  its 
direction  on  the  blackboard.  Read  the  dynamometers  and  record  the 


38 


A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


magnitude  of  each  force  on  the  corresponding  line.     Adopt  a  conven* 
ient  scale  and  lay  off  each  force  along  its  line  of  direction,  measuring 

from  O.  Using  the  ob- 
lique lines  as  sides,  con- 
struct the  parallelogram. 
Measure  the  diagonal  OR 
and  write  its  value  in 
force  units  upon  it.  A 
comparison  will  show  that 
the  force  represented  by 
OR  is  equal  to  W  and 
might,  therefore,  be  sub- 
stituted for  the  two 
components  and  would 
produce  the  same  effect. 

44.  Equilibrant  and 
Resultant.  --  The 
force  W  is  said  to  hold  the  component  forces  in  equilib- 
rium, arid  is  therefore  called  the  equilibrant  (pronounced 
<? quili' brant) .  From  the  definitions  given  of  resultant 
and  equilibrant,  we  see  that  they  are  necessarily  equal 
in  magnitude  but  opposite  in  direction. 


FIG.  19,  —  Principle  of  the  Parallelogram  of 
Forces  Illustrated. 


EXERCISES 

1.  Represent  by  a  diagram  the  resultant  of  two  forces  of  15  dynes 
and  25  dynes  acting  (1)  in  the  same  direction  and  (2)  in  opposite 
directions  from  the  same  point. 

2.  Find  the  magnitude  and  the  direction  of  the  resultant  of  two 
forces,  3  Ib.  acting  north  and  4  Ib.  acting  west,  applied  at  the  same 
point. 

3.  A  ball  is  acted  upon  simultaneously  by  two  forces,  one  of  10  kg. 
directed  upward,  the  other  of  25  kg.  directed  east  along  a  horizontal 
line.     Find  the  resultant  in  both  magnitude  and  direction. 

4.  The  angle  between  a  force  of  50  dynes  and  one  of  30  dynes  is 
60°.     Find  the  resultant  and  the  equilibrant  in  both  magnitude  and 
direction. 

5.  The  angle  between  two  equal  forces  of  40  Ib.  each  is  120°.     Find 
the  resultant. 


LAWS  OF  MOTION  — FORCE  39 

6.  A  boat  is  pulled  by  two  ropes  making  an  angle  of  30°.     If  one 
force  is  10  lb.  and  the  other  20  lb.,  what  is  the  resultant? 

7.  A  weight  is  suspended  by  two  cords  applied  at  the  same  point 
and  each  making  an  angle  of  30°  with  a  vertical  line.     If  the  tension 
in  each  is  25  lb.,  what  is  the  weight  supported  ? 

Ans.  43.3  IK 

3.  MOMENTS  OF  FORCE  —  PARALLEL  FORCES 

45.  The  Moment  of  a  Force.  —  It  can  often  be  observed 
that  when  a  mechanic  wishes  to  loosen  a  nut  that  is  diffi- 
cult to  start,  he  uses  a  wrench  with  a  long  handle.  For 
those  that  start  easily,  he  uses  a  short-handled  wrench. 
The  results  prove  that  the  effectiveness  of  a  force  in  pro- 
ducing rotation  against  a  resistance  is  greater  as  the  ap- 
plied force  is  farther  from  the  point  about  which  rotation 
takes  place. 

The  effectiveness  of  a  force  in  producing  a  rotation  is  called 
the  moment  of  the  force.  The  moment  of  a  force  depends 
upon  two  quantities:  (1)  the  magnitude  of  the  force  and 
(2)  the  perpendicular  distance  from  the  point  about  which 
the  rotation  takes  place  to  the  line  representing  the  direction 
of  the  force.  The  moment  is  measured  by  the  product  of 
these  two  factors.  The  following  experiment  will  make 
the  matter  clear: 

Fasten  one  end  of  a  light  wooden  bar  to  the  table  top  by  means  of 
a  nail  at  0,  Fig.  20.  Let  a  force,  which 
may  be  measured  by  a  dynamometer  (see 
Fig.  6),  pull  upon  the  bar  at  A,  and  an- 
other at  B,  as  shown.  Measure  both  forces 
and  the  distances  AO  and  BO.  The  prod- 
uct of  the  force  applied  at  A  multiplied 
by  the  distance  A  0  will  be  found  equal  to 
the  product  of  the  other  force  multiplied 
by  the  distance  BO. 

LJ0. 

It  is  clear  from  this  experiment 

,  FIG.   20.  — The  Equality  ol 

that  a  force  that  tends  to  turn  a  body        Moments  illustrated. 


40  A  HIGH   SCHOOL  COURSE  IN  PHYSICS 

to  the  right  can  be  balanced  by  another  of  the  same  moment 
that  tends  to  produce  rotation  to  the  left. 

46.  Parallel  Forces.  —  Objects  are  frequently  supported 
by  two  or  more  upward  forces  acting  at  different  points, 
thus  forming  a  system  of  parallel  forces.  For  example, 
two  men  may  support  a  heavy  beam  or  carry  a  loaded 
bucket  on  a  bar  between  them.  A  bridge  is  supported  by 
the  upward  pressures  of  the  piers  at  the  ends.  The  prin- 
ciple of  moments  given  in  the  preceding  section  is  of 
service  in  determining  the  resultant  of  such  forces  as 
the  following  experiment  illustrates: 

Select  a  bar  of  wood  4  or  5  ft.  in  length,  of  uniform  width  and 
uniform  thickness.  A  pine  board  about  4  in.  in  width  is  convenient. 

Place  hooks  at  several  points 
along  one  edge,  as  shown  in 
Fig.  21,  but  place  one  hook 
so  that  the  bar  wilMSalanee 
well  when  hung  from  that 
point.  Call  this  point  C. 

W  Suspend  the  bar  from  a  dy- 

namometer at  C  and  ascer- 
FIG.  21.  -  Law  of  Parallel  Forces  tain  the  wei^ht  of  it.     This 

Illustrated.  .„  .      ,.  ,  ,, 

will  be  the  value  of  the  re- 
sultant in  every  case.  Now  release  the  bar  at  C  and  attach  dynamom- 
eters at  two  points,  say  A  and  B,  and  ascertain  the  forces  required 
to  support  the  bar.  Designating  these  forces  by  F  and  F',  it  will  be 
found  that  in  every  case  the  sum  of  the  two  components  is  equal  to 
the  weight  W  of  the  bar  ;  or,  F  +  F  =  W.  Furthermore,  the  moment 
of  the  force  F  about  the  point  Bm  (i.e.  F  x  AB)  will  be  found  equal  to 
the  moment  of  the  weight  of  the  blar  about  the  same  point  (i.e.  W  X 
CB)  ;  or, 

F  x  AB  =  W  x  CB.  (3) 

The  moment  of  the  component  F1  about  the  point  A  will  be  found 
equal  to  the  moment  of  W  about  the  same  point.  Hence  we  may  write 

F  x  AB  =  W  x  AC.  (4) 

The  laws  of  parallel  forces  may  therefore  be  stated  as 
follows : 


LAWS  OF  MOTION  — FORCE  41 

1.  The  resultant  of  two  parallel  forces  acting  in  the,  same 
direction  at  different  points  on  a  body  is  equal  to  their  sum, 
and  has  the  same  direction  as  the  components. 

2.  The  moment  of  one  of  the  components  about  the  point 
of  application  of  the  other  is  equal  and,  opposite  to  the  moment 
of  the  supported  weight  about  the  same  point. 

EXAMPLE.  —  Two  men,  A  and  B,  carry  a  bucket  \veighing  100  Ib. 
on  a  bar  10  ft.  long.  If  the  bucket  is  4  ft.  from  A,  how  much  force 
is  exerted  by  each? 

SOLUTION.  —  The  moment  of  the  force  F  exerted  by  A  about  the 
opposite  end  of  the  bar  is  10  x  F,  and  the  moment  of  the  weight  about 
the  same  point  is  100  x  (10  -  4),  or  600.  Hence  10  x  F  =  600 ;  whence 
F  =  60  Ib.  Let  F'  be  the  force  exerted  by  B.  Then  considering  the 
moments  about  the  other  end  of  the  bar,  we  have  10  x  F'  =  100  x  4 ; 
whence  F  =  40  Ib.  Therefore  A  exerts  60  Ib.  and  B  40  Ib. 

,x 

47.  The  Couple.  —  When  two  equal  parallel  forces  act 
upon  a  body  along  different  lines  and  in  opposite  direc- 
tions, as  shown  in  Fig.  22,  they  have  no 
resultant;  that  is,  no  single  force  will  have 
the  same  effect  as  the  two  components  act- 
ing jointly.  A  combination  of  this  kind  is 
called  a  cougle.  The  tendency  of  a  couple  is 
always  to  rotate  the  body  on  which  it  acts. 
This  tendency  is  measured  by  the  moment  of 
the  couple,  which  is  the  product  of  one  of 
the  forces  multiplied  by  the  perpendicular  FIG.  22.  — The 
distance  AB  between  the  two  forces.  This  Couple, 
distance  is  the  arm  of  the  couple.  The  equilibrium  of  the 
body  acted  upon  can  be  maintained  only  by  the  applica- 
tion of  another  couple  of  equal  moment  acting  in  the  op- 
posite direction. 

A  small  magnet  placed  on  a  floating  cork  is  rotated  by 
the  couple  formed  by  the  northward-acting  force  at  one 
end  and  the  equal  southward-acting  force  at  the  other. 


42  A  HIGH   SCHOOL  COURSE  IN  PHYSICS 

EXERCISES 

NOTE.  —  The  student  should  first  draw  a'diagram  representing  the 
conditions  of  the  problem  to  be  solved  and  then  apply  the  general 
results  deduced  from  the  experiment  m  §  46. 

1.  A  uniform  bar  of  wood  weighing  12  kg.  is  120  cm.  long;  two 
hooks  are  placed  on  opposite  sides  of  the  center  at  distances  of  40  cm. 
and  20  cm.  respectively.     What  forces  applied   to   the  hooks  will 
support  the  bar?  Ans.  4  kg.  and  8  kg. 

2.  In  order  to  support  the  bar  in  Exer.  1,  what  forces  applied  at 
points  respectively  15  cm.  and  15  cm.  from  the  ends  of  the  bar  will 
be  required  ?  A  ns.  9  kg.  and  3  kg. 

3.  A  beam  of  uniform  size  is  60  ft.  long  and  weighs  800  lb.;  a 
man  at  one  end  supports  200  lb.     Find  the  magnitude  and  point  of 
application  of  the  other  required  force. 

4.  Two  parallel  forces  of  30  g.  and  70  g.  are  applied  at  the  ends  of 
a  bar  1  m.  long.     Find  what  weight  will  be  supported  and  its  location 
on  the  bar,  neglecting  the  weight  of  the  bar  itself. 

5.  A  boy  and  a  man  are  carrying  a  weight  of  150  lb.  on  a  bar  10 
ft.  in  length.     If  the  forces  are  applied  at  the  ends  of  the  bar,  where 
must  the  load  be  placed  in  order  that  the  boy  may  have  to  carry  only 
50  lb.  ? 

6.  Draw  a  diagram  showing  a  method  for  attaching  three  horses 
to  a  load  so  that  they  must  pull  equally. 

4.  RESOLUTION  OF  FORCES 

48.  Resolution  of  Forces.  —  In  many  cases  it  becomes 
desirable  to  find  the  effect  of  a  force  in  some  direction 
other  than  that  in  which  it  acts.  For  example,  a  car  on  a 

track  running  east  and  west,  Fig. 
23,  is  acted  upon  by  a   force  AB 
directed    northeast.      The    given 
force  has  two  effects  :   It  produces 
(1)  a  tendency  to  move   the    car 
FIG.  23.— Force  Resolved       east,  and  (2)   a  pressure  against 
into  Two  Components.         the  rails  toward  the  north.     It  is 

plain  that  two  forces,  one  directed   east   and   the  other 
north,  might  have  the   same   effect  as  the   single   force 


LAWS  OF  MOTION  — FORCE 


43 


directed  northeast.  In  order  to  determine  these  two 
forces,  the  given  force  is  represented  by  the  line  AB,  from 
whose  extremities,  A  and  .#,  lines  are  drawn  completing 
the  rectangle,  as  shown  in  the  figure.  AC  is  called  the 
effective  component,  since  it  acts  in  the  direction  in  which 
the  car  can  move ;  and  AD  is  designated  as  the  non~ 
effective  component,  since  it  contributes  nothing  to  the 
production  of  motion. 

A  given  force  may  be  resolved  into  two  components  whose 
directions  are  given  by  making  the  line  of  force  the  diagonal 
of  a  parallelogram  whose  sides  are  drawn  from  the  point  of 
application  of  the  force  in  the  directions  required  for  the 
components. 

49.  The  Sailboat  and  the  Aeroplane.  —  The  principle  of  the 
resolution  of  forces  explained  above  is  readily  applied  to  the  opera- 
tion of  a  sailboat.  Let  the  boat  be  headed  north  while  the  wind  blows 
from  the  east.  Now  the  pressure  of  the  wind  AC  on  the  sail  SS', 
(1),  Fig.  24  can  be  resolved  into  A  B,  perpendicular  to  the  sail,  and  a 


FIG.  24. 

second  component  AD,  parallel  to  the  sail,  the  latter  of  which  is  non- 
effective.  Force  AB  is  the  effective  pressure  on  the  sail.  If  the 
vessel  were  round,  it  would  move  in  the  direction  of  AB.  Now  let 
AB  be  resolved  as  shown  in  (2),  Fig.  24  into  A  E  acting  parallel  to  the 
keel  and  AF  acting  perpendicular  to  it.  The  former  component 


44 


A  HIGH   SCHOOL  COURSE  IN  PHYSICS 


Direction  of  Flight 


moves  the  vessel  forward,  while  the  component  AE  is  rendered  non 
effective  by  the  deep  keel  of  the  boat. 

In  the  case  of  the  aeroplane,  which  is  a  recent  invention,  huge 
planes  or  sails,  AE  and  CD,  shown  in  the  sectional  view,  Fig.  25,  are 
attached  firmly  to  a  light  frame,  upon  which  is  mounted  a  powerful 
gasolene  motor  (§  271).  The  planes  are  slightly  oblique,  as  shown. 
The  power  furnished  by  the  motor  turns  a  propeller  whose  office  it  is 
to  drive  the  aeroplane  rapidly  forward.  When  the  aeroplane  moves 

forward  to  the  left,  it  is  as  though  a  strong 
wind  were  blowing  toward  the  right 
against  the  planes,  as  shown  by  the  dotted 
lines.  As  in  the  case  of  the  sailboat 
a  pressure  is  produced  at  right  angles 
to  the  planes  AE  and  CD.  Represent- 
ing this  force  by  the  line  EF  and  re- 
solving it  into  two  components,  we  find 
the  lifting  force  EG,  and  the  component 
EH,  which  tends  to  resist  the  forward  mo- 
tion of  the  aeroplane.  Smaller  planes  whose  positions  can  be  changed 
by  the  operator,  are  used  in  steering. 


FIG.  25. 


EXERCISES 

1.  If  the  force  represented  by  the  line  AB,  Fig.  23,  is  1000  lb.,  and 
the  angle  BA  C,  60°,  find  the  components  A  C  and  AD. 

2.  Resolve  a  force  of  2000  dynes  into  two  components  making 
angles  of  30°  and  60°  with  the  given  force. 

3.  A  weight  of  50  kg.  is  suspended  by  two  cords  making  angles  of 
30°  and  60°  respectively  with  the  vertical.     Find  the  force  exerted  by 
each  cord.  Ans.   25  kg.  and  43.3  kg. 

4.  If  the  mass  of  the  car  in  Exer.  1  is  20,000  lb.,  and  the  resistance 
offered  by  the  rails  may  be  neglected,  what  is  the  acceleration  of  the 
car? 

SUGGESTION.  —  Reduce  the  effective  component  to  poundals  and 
apply  equation  (2),  §  36.  Ans.   0.161  ft.  per  sec.  per  sec. 


5.   CURVILINEAR  MOTION 

50.  Uniform  Circular  Motion.  —  It  is  a  well-known  fact 
that  a  ball  attached  to  one  end  of  a  cord  and  whirled 
about  the  hand  exerts  a  pulling  force  against  the  hand 


LAWS  OF  MOTION  —  FORCE  45 

along  the  cord.  This  takes  place  because  of  the  tendency 
of  the  ball  to  move  in  a  straight  line  according  to  New- 
ton's First  Law  of  Motion*  In  order,  therefore,  to  confine 
the  ball  to  a  circular  path,  a  continual  force  toward  the 
center  must  be  maintained.  If  this  force  is  removed  by 
the  breaking  of  the  cord,  the  ball  will  leave  its  circular 
path  along  a  tangent.  The  force  that  continually  deflects 
a  moving  body  from  a  straight  line,  compelling  it  to  follow 
a  curve,  is  called  centripetal  force.  If  the  motion  of  a 
body  along  the  circumference  of  a  circle  is  uniform,  the 
centripetal  force  is  constant. 

The  name  "  centrifugal  "  force  is  often  applied  to  the  reaction  of 
the  moving  body  upon  the  fixed  center.  This  reaction  gives  one  the 
erroneous  impression  that  the  body  would  fly  away  from  the  center 
along  a  radius  if  the  centripetal  force  should  cease  acting.  If,  how- 
ever, we  watch  the  course  taken  by  water  or  mud  as  it  leaves  a  revolv- 
ing wheel,  we  readily  observe  that  it  moves  along  the  tangent  to  the 
wheel  at  the  point  where  it  is  set  free. 

51.  Centripetal  Acceleration.  —  Since  a  force-produces 
a  change  in  momentum  in  the  direction  of  the  force, 
according  to  Newton's  Second  Law  (§  35),  a  constant  cen- 
tripetal force  produces  a  constant  change  in  the  momen- 
tum of  a  body,  which  has  uniform  circular  motion,  toward 
the  center.  Since  the  mass  is  constant,  the  acceleration 
is  constant  and  directed  toward  the  center.  If  v  is  the 
velocity  with  which  a  body  is  moving  in  a  circular  path, 
and  r  the  radius  of  the  circle,  the  centripetal  acceleration 
a  is  represented  as  follows  i1 

a  =  -  (5) 


1  Let  a  body  m  be  moving  around  the  circle  whose  center  is  o,  with  a 
uniform  velocity  v.  Let  it  move  from  TO  to  c  in  the  very  short  time  of  t 
seconds.  Then  the  distance  me  will  be  equal  to  vt  (Eq.  1,  p.  14).  If  the 
time  is  small,  the  arc  me  is  practically  equal  to  the  chord  me.  On  com- 


46 


A  HIGH   SCHOOL  COURSE  IN  PHYSICS 


Thus,  if  the  moving  body  is  at  the  point  TO,  Fig.  26,  its  tendency  is 
to  continue  in  the  direction  mb  in  accordance  with  the  First  Law  of 

Motion.  But,  on  account  of  the 
cord,  it  is  compelled  to  keep  the 
same  distance  from  the  center  o  and, 
consequently,  is  deflected  from  b  to  c. 
Again,  at  c,  as  at  every  point,  the 
body  tends  to  follow  the  tangent  ce, 
but  is  compelled  to  take  an  inter- 
mediate path  along  the  circumfer- 
ence to  /.  If  the  deflecting  force 
is  removed  at  the  time  the  body 
reaches  /,  it  continues  to  move  in 
the  direction  of  its  motion  at  that 
FIG.  26.  —  A  Body  ra  Having  Circu-  point ;  that  is,  along  the  line  fg, 

lar  Motion  Tends  to  Follow  the    which  is  tangent  to  the  circle  at  /. 

Tangent  to  its  Path.  T.    .     ,,  ,  ,,  , . 

It  is  the  component  of  the  motion 

ma  produced  by  the  centripetal  force  acting  along  mo  whose  accelera- 
tion is  represented  in  equation  (5). 

52.  Equation  of  Centripetal  Force.  —  A  force  is  equal  to 
the  product  of  the  mass  of  the  body  upon  which  it  acts 
and  the  acceleration  that  it  produces  (§  36).  Hence 
in  circular  motion  the  centripetal  force  is  the  product  of 


2 

the  mass  m  and  the  centripetal  acceleration  — . 

r 


Therefore 


the  value  of  the   centripetal  force   may  be  expressed  as 
follows : 

pleting  the  small  rectangle  m&ca,  we  have,  since  ca  is  a  perpendicular 
dropped  upon  the  hypotenuse  "of  the  right  triangle  mch, 

me*  =  ma  x  ~mh.  (1) 

Now  the  distance  the  mass  m  is  drawn  toward  the  center  by  the  con- 
stant centripetal  force  in  the  time  t  is  be  and  equals  ma.  Since  the  motion 
toward  the  center  is  uniformly  accelerated, 

!nc  =  ma  =  $  at2  (Eq.  3,  p.  18).  (2) 

Therefore,  by  substituting  the  values  we  =  vt  and  ~m~a  =  |  otf2  in  (1),  we 
have  vW  =  \  at2  x  2  r,  where  r  is  the  radius  of  the  circle.     From  this 

equation  v2  =  ar,  whence  a  =  — . 


LAWS   OF  MOTION  — FORCE  47 

TH  v 
Centripetal  |orce  =  — — .  (6) 

It  should  be  observed  that  this  equation  gives  the  force 
in  absolute  units  only ;  i.e.  in  dynes,  when  C.  G.  S.  units 
are  substituted,  and  in  poundah,  when  F.  P.  S.  units  are 
employed. 

53.  Illustrations  of  Circular  Motion.  —  Many  examples 
of  circular  motion  present  themselves  in  everyday  life. 
The  bicycle  rider  must  carefully  govern  his  speed  as  he 
turns  a  corner  on  a  slippery  pavement  lest  the  force  re- 
quired to  change  the  direction  of  motion  be  too  great  and 
the  wheels  slip  sidewise.  In  the  modern  cream  separators 
the  denser  portions  of  the  milk  are  forced  to  the  outside 
of  a  rapidly  revolving  bowl,  while  the  lighter  cream  re- 
mains near  the  center  and  is  forced  out  along  the  axis. 
Honey  is  extracted  by  rapidly  whirling  the  uncapped  comb 
in  a  machine.  Centrifugal  driers  are  used  in  laundries 
and  factories  for  removing  water  from  clothing,  wool,  etc. 
In  the  "  loop  the  loop  "  apparatus  a  car  rides  safely  along 
a  track  within  a  large  vertical  circle,  its  own  tendency  to 
follow  a  tangent  keeping  it  pressed  firmly  against  the 
rails. 

The  motion  of  the  bodies  of  the  solar  system  illustrates 
the  action  of  centripetal  force  on  the  grandest  scale.  The 
earth,  for  example,  having  an  initial  motion,  tends  to  move 
in  a  straight  line.  However,  the  attraction  of  the  sun, 
like  a  tense  cord,  holds  it  in  its  orbit.  If  this  force  should 
cease,  the  earth  would  at  once  move  away  into  space  along 
a  tangent  to  its  orbit.  On  the  other  hand,  if  it  were  not 
for  the  earth's  motion  along  the  curve,  it  would  be  drawn 
with  accelerated  motion  into  the  sun. 

The  spheroidal  shape  of  the  earth  is  supposed  to  be  due 
to  the  tendency  of  matter  to  withdraw  from  the  axis  of 


48  A  HIGH   SCHOOL  COURSE  IN  PHYSICS 

rotation.     This  tendency  causes  bodies  to  weigh  about  ^Jg 
less  at  the  equator  than  at  the  poles. 

When  the  centripetal  force  is  not  sufficient  to  keep  the 
parts  of  a  revolving  body  in  the  required  circular  paths, 
serious  results  often  follow.  This  is  the  case  of  bursting 
fly  wheels  and  emery  wheels  in  mills  and  factories. 

EXERCISES 

1.  Explain  why  water  will  not  fall  from  a  pail  whirled  at  arm's 
length  in  a  vertical  circle. 

2.  How  is  the  overturning  of  a  car  prevented,  as  it  rapidly  turns  a 
curve  ? 

3.  Does  the  rotation  of  the  earth  affect  the  weight  of  bodies  in  this 
latitude  ? 

4.  Account  for  the  fact  that  the  moon  moves  in  an  orbit  around 
the  earth. 

5.  What  keeps  the  earth  in  rotation  on  its  axis  ? 

6.  Show  by  equation  (5)  that  increasing  the  rate  of  rotation  of  the 
earth  seventeen  fold  would  cause  bodies  at  the  equator  to  "  lose  "  their 
entire  weight. 

7.  A  body  whose  mass  is  50  g.  moves  in  a  circle  whose  radius  is 
40  cm.  with  a  velocity  of  20  cm. /sec.     What  is  the  required  centripe- 
tal force?  v 

8.  A  stone  leaves  a  sling  with  a  velocity  of  50  ft.  per  second.     If 
the  mass  of  the  stone  is  2  oz.  and  the  radius  of  the  circle  4  ft.,  what 
was  the  pull  exerted  on  the  cords  of  the  sling  ? 

Ans.  78.125  poundals. 

SUMMARY 

1.  The  momentum  of  a  body  is  measured  by  the  prod- 
uct of  its  mass  and  velocity.     It  is  represented  by  the 
expression  mv  (§  33). 

2.  The  term  force  is  the  name  given  to  the  cause  that 
produces   acceleration,  retardation,    or   a   change   in   the 
direction  of  the  motion  of  a  body  (§  34). 

3.  Force  is   measured   by   the   change   in   momentum 
produced  per  second.     The  C.  G.  S.  unit  of  force  is  the 


LAWS   OF  MOTION  — FORCE  49 

dyne.  The  dyne  is  that  force  which,  acting  uniformly 
for  one  second,  imparts  one  C.  G.  S.  unit  of  momentum 
(§  35). 

4.  The  equation  of  force  is/=  — ,  or/=  ma  (§  36). 

t 

5.  The  absolute  units  of  force  are  the  dyne  and  poundal ; 
the  gravitational  units  are  the  gram-weight  and  pound-weight, 
etc.     In  everyday  use  the  gravitational  units  are  called 
simply  the  "gram  "  and  "pound  "  (§  37). 

6.  A  given  force  produces  its  own  effect,  whether  act- 
ing alone  or  conjointly  with  other  forces  (§  38). 

7.  To  every  action  there  is  always  an  equal  and  oppo- 
site reaction;  or,  in  other  words,  for  every  push  or  pull  of 
one  body  upon  a  second  body  there  is  always  an  equal  pull 
or  push  of  the  second  body  upon  the  first  (§  39). 

8.  The  characteristics  of  a  force  are  its  point  of  appli- 
cation, direction,  and  magnitude.     Forces  are  represented  by 
straight  lines  of  suitable  length  and  direction  and  may  be 
compounded  in  the  same  manner  as  motions  and  velocities 
(§  40). 

9.  The  resultant  of  two  or  more  forces  acting  in  the  same 
direction  along  a  straight  line  is  equal  to  their  sum;  but 
when  two  forces  act  in  opposite  directions  in  the  same  line, 
their  resultant  is  equal  to  their  difference  and  has  the  di- 
rection of  the  greater  force  (§  42). 

10.  The  resultant  of  two  forces  acting  at  an  angle  is  rep- 
resented by,  the  diagonal  of  a  parallelogram  constructed  on 
the  lines  which  represent  the  component  forces.     This  law 
is  universally  known  as  the  Principle  of  the  Parallelogram 
of  Forces  (§  43). 

11.  The  moment  of  a  force  about  a  point  is  the  effective- 
ness of  the  force  in  producing  a  rotation.     It  is  measured 
by  the  product  of  the  magnitude  of  the  force  and  the  per- 

5 


50    '         A  HIGH   SCHOOL  COURSE  IN  PHYSICS 

pendicular  distance  from  the  point  to  the  line  of  direction 
of  the  force  (§  45). 

12.  Any  two  parallel  forces  acting  upward  will  support 
a  weight   equal   to   their  sum,  and  the    moment  of   one 
component  about  the  point  of  application  of  the  other  is 
equal  and  opposite  to  the  moment  of  the  supported  weight 
about  the  same  point  (§  46). 

13.  A  system  of  two  equal  and  opposite  parallel  forces 
acting  along  different  lines  is  called  a  couple.     The  moment 
of  a  couple  is  the  product  of  one  of  the  forces  multiplied  by 
the  distance  between  the  two  forces.     A  couple  can  be 
balanced  only  by  another  couple  acting  in  the  opposite 
direction  and  having  an  equal  moment  (§  47). 

14.  A  force  may  be  resolved  into  two  components  by 
making  it  the  diagonal  of  a  parallelogram  whose  sides  are 
drawn  in  the  directions  required  for  the  components  (§  48.) 

15.  When  a  body  has  curvilinear  motion,  a  force  is  re- 
quired to  deflect  the  body  continually  from  a  straight  line. 
This  is  called  centripetal  force.     The  equation  of  centripetal 


force  is/  =          (§52). 


CHAPTER   IV 
WORK  AND  ENERGY 

1.    DEFINITION   AND    UNITS   OF   WORK 

54.  Work.  —  The  use  of  the  expression  "  to  do  work  " 
is  restricted  in  the  study  of  mechanics  to  cases  in  which 
a  force  produces  motion  in  the  body  upon  which  it  acts. 
For  example,  attempting  to  lift  a  stone  from  the  ground 
without  succeeding  in  moving  it  is  not  doing  work  in  the 
scientific  sense ;  but  lifting  the  stone  to  a  higher  position 
implies  that  work  is  being  done  upon  it.     Similarly,  bend- 
ing a  bow  is  doing  work,  but  holding  it  in  a  bent  condi- 
tion is  not.     Lifting  a  weight   involves  the  process  of 
doing  work,  but  simply  supporting  it  does  not.     Work 
is  done  when  the  spring  of  a  clock  is  wound,  or  a  body  is 
moved  along  upon  a  table. 

An  important  case  in  which  work  is  done  is  that  in 
which  a  freely  moving  body  is  given  acceleration.  We 
have  already  found  (§  34)  that  an  increase  in  the  velocity 
of  a  body  requires  the  action  of  a  force.  Furthermore, 
the  tendency  of  the  force  is  to  produce  motion  in  the 
direction  in  which  the  force  acts.  "  Hence,  work  is  done 
by  exploding  powder  when  it  projects  a  bullet  from  a  gun, 
or  by  gravity  when  a  body  is  allowed  to  fall  to  the  earth. 

55.  Elements  Involved  in  Work.  —  In  each  of  the  ex- 
amples given  in  the  preceding  section,  it  will  be  observed 
that  two  quantities  are  involved  in  the  process  of  doing 
work.     These  are  (1)  the  acting  force  and  (2)  the  distance 
through  which  the  force  continues  to  act,  sometimes  called 
the  displacement.     Work  is  directly  proportional  to  the  force 

61 


52  A  HIGH   SCHOOL   COURSE   IN    PHYSICS 

and  the  distance  through  which  the  force  acts,  and  is  meas- 
ured by  their  product.     Thus 

Work  =  force  x  displacement. 

Or,  if /represents  the  force,  d  the  distance  through  which 
the  force  acts. 

Work  =  fd.  (i) 

56.  Units  of  Work.  —  Since  work  is  measured  by  the 
product  of  the  force  that  acts  upon  a  body  and  the  dis- 
tance the  body  is  moved  in  the  direction  of  the  force,  a 
unit  of  work  is  done  when  a  unit  of  force  acts  through  a  unit 
of  distance.  For  every  unit  of  force  (§§  36  and  37)  there 
is  a  corresponding  unit  of  work.  The  most  important, 
however,  is  the  C.  G  .S.  unit  which  is  called  the  erg.  The 
erg  is  the  work  done  when  a  force  of  one  dyne  acts  through  a 
distance  of  one  centimeter.  The  erg  is  also  called  the  dyne- 
centimeter. 

The  F.  P.  S.  unit  of  work  is  the  foot-poundal,  which  is  the  work  done 
when  a  poundal  of  force  acts  through  a  distance  of  one  foot.  In  the 
gravitational  system  two  units  are  frequently  used.  The  kilogram- 
meter  is  the  work  done  when  a  force  of  one  kilogram  (§  37)  acts 
through  a  distance  of  one  meter.  The  foot-pound  is  the  work  done 
when  a  force  of  one  pound  acts  through  a  distance  of  one  foot. 

The  following  table  is  given  to  show  the  use  of  equa- 
tion (l)  in  the  computation  of  work  in  the  different 
systems  : 


ABSOLUTE  SYSTEM 

/-(in  dynes)        x  d  (in  centimeters)  =  Work  (in  ergs) . 

/  (in  poundals)  x  d  (in  feet)  =  Work  (in  foot-poundals). 


GRAVITATIONAL  SYSTEM 

/(in  kilograms)  x  d  (in  meters)      =  Work  (in  kilogram-meters), 
/(in  pounds)       x  d  (in  feet)  =  Work  (in  foot-pounds). 


WORK  AND   ENERGY  53 

EXAMPLE.  —  A  mass  of  50  g.  requires  a  force  of  10  g.  to  overcome 
the  friction  and  move  the  body  at  a  uniform  rate  along  a  horizontal 
table.  Find  the  work  done  when  the  mass  is  moved  horizontally 
25  cm.  Find  also  the  work  done  when  the  mass  is  lifted  25  cm. 

SOLUTION.  —  Since  the  horizontal  force  is  10  g.  or  9800  dynes 
(§  37).  and  the  distance  through  which  the  force  acts  is  25  cm.,  the 
work  performed  is  10  x  25,  or  250  gram-centimeters.  Measured  in 
ergs,  the  work  is  9800  x  25,  or  245,000  ergs. 

The  work  performed  in  lifting  the  mass  is  50  x  25,  or  1250  gram- 
centimeters.  Measured  in  ergs,  the  work  is  49,000  x  25,  or  1,225,000, 
ergs. 

The  numerical  relation  between  the  various  units  of 
work  is  shown  in  the  following  table  : 

1  dyne-centimeter  equals  1  erg. 

1  kilogram-meter  (kg-m.)  equals  98,000,000  ergs. 
1  foot-pound  equals  13,550,000  ergs. 

1  foot-poundal  equals      421,390  ergs. 

2.    ACTIVITY,   OR   RATE   OF   WORK 

57.  Activity.  —  The  value  of  any  agent  employed  in  do- 
ing work  will  depend  upon  the  amount  of  work  it  is  able 
to  perform  in  a  certain  time.     Some  agents  work  slowly, 
others   rapidly.     For  example,  a  man  can  lift  a  certain 
number  of  bricks  to  the  top  of  a  building  in  an  hour ;  a 
horse  attached  to  a  suitable  hoisting  mechanism  can  lift  a 
greater  number  in  the  same  time  ;  and  an  engine  can  lift 
the  bricks  as  fast  as  several  horses.     Working  agents  are 
therefore  said  to  differ  in  activity,  or  power.     Activity  is 
the  rate  of  doing  work,  and  is  found  by  dividing  the  work 
performed  by  the  time  consumed  in  the  process. 

58.  Units  of  Activity.  — The  unit  of  activity  or  power 
commonly  used  is  the  horse  power   (abbreviated  H.P.). 
The  horse  power  is  the  rate  of  doing  work  equal  to  550  foot- 
pounds per  second.     The  activity  of  an  agent  that  is  able 
to  perform  550  foot-pounds  of  work  per  second  is  one  horse 


54  A  HIGH   SCHOOL  COURSE   IN  PHYSICS 

power.  The  unit  of  power  in  the  C.  G.  S.  system  is  the 
watt,1  which  is  equivalent  to  the  work  done  at  the  rate  of 
107  ergs  per  second.  One  horse  power  equals  746  watts, 
or  746  x  107  ergs  per  second. 

EXERCISES 

1.  Calculate  the  work  done  by  a  force  of  25  dynes  acting  through 
a  distance  of  120  cm. 

2.  Express  in  ergs  and  gram-centimeters  the  work  done  in  lifting 
a  mass  of  5  g.  through  a  vertical  height  of  100  cm. 

3.  A  horse  has  to  exert  an  average  force  of  200  Ib.  in  moving  a 
loaded  cart  a  distance  of  a  mile.     Find  the  amount  of  work  done. 

4.  What  amount  of  work  is  done  when  one  cubic  meter  of  water 
is  elevated  to  a  height  of  10  m.? 

5.  How  much  work  is  done  per  second  by  an  engine  that  in  one 
hour  lifts  10,000  bricks  each  weighing  4  Ib.  to  the  top  of  a  building  50 
ft.  in  height?     Find  the  necessary  horse  power. 

6.  A  man  shovels  3  T.  of  coal  from  a  wagon  box  into  a  bin  6  ft. 
above  the  coal  in  the  wagon.     How  much  work  is  involved  in  the 
process? 

7.  What  must  be  the  power  of  an  engine  that  hoists  50  T.  of  ore 
per  hour  from  a  mine  300  ft.  deep? 

8.  A  pumping  engine  is  capable  of  raising  300  cu.  ft.  of  water 
every  minute  from  a  mine  132  ft.  in  depth.     If  a  cubic  foot  of  water 
weighs  62.5  Ib.,  what  must  be  the  power  of  the  engine  ? 

9.  How  long  will  it  take  a  3-H.  P.  engine  to  elevate  5000  bu.  of 
wheat  50  ft.?    (A  bushel  of  wheat  weighs  60  Ib.) 

10.  A  train  is  moving  with  a  velocity  of  30  mi.  per  hour.     If  the 
resistance  to  the  motion  is  1500  Ib.,  calculate  the  power  utilized. 

11.  The  motors  of  an  electric  car  can  develop  200  H.  P.     With 
what  velocity  can  the  car  run  against  a  uniform  resistance  of  2200  Ib.  ? 

3.    POTENTIAL   AND   KINETIC   ENERGY 

59.  Energy.  —  In  each  of  the  cases  selected  in  §  54  to 
illustrate  the  process  of  doing  work,  some  agent  capable  of 
doing  work  was  assumed  to  be  acting.  The  stone,  for 

1  So  called  in  honor  of  James  Watt  (1736-1819),  the  inventor  of  the 
steam  engine. 


WORK  AND  ENERGY  55 

example,  was  supposed  to  be  lifted  by  this  agent,  which 
may  have  been  an  engine,  a  person,  a  horse,  or  any  other 
working  medium.  In  order  to  be  able  to  perform  work, 
an  agent  must  possess  energy.  The  energy  of  a  body  is 
its  capacity  for  doing  work,  or  its  ability  to  do  work. 

If  we  examine  a  body  upon  which  work  has  been  done, 
—  any  lifted  mass,  for  instance,  —  we  discover  that  the 
lifting  process  has  invested  the  mass  with  the  ability  to  do 
work  upon  some  other  body.  The  lifted  body  may  be 
attached  by  a  cord  to  the  proper  mechanism  and  allowed 
to  fall  back  to  its  original  position ;  but  during  its  fall  it 
may  turn  wheels,  lift  another  body,  bend  a  bow,  wind  a 
spring,  or  do  work  in  some  other  manner.  Thus  in  doing 
work  the  falling  body  gives  energy  or  working  ability  to  that 
upon  which  the  work  is  done.  Similarly,  a  hammer  by 
virtue  of  the  velocity  given  it  by  the  mechanic  possesses 
the  capacity  for  doing  work;  This  is  manifested  by  the 
fact  that  it  drives  the jiail  in  opposition  to  the  resistance 
offered  by  the  wood. 

60.  Potential  Energy.  —  The  lifted  weight  in  the  illus- 
tration used  in  the  preceding,  section  possesses  energy  be- 
cause of  its  elevated  position.  ~"*A  bent  bow  has  the  ability 
to  throw  an  arrow  because  of  the  fact  that  its  form  has 
been  changed  by  some  agent.  The  spring  of  a  watch  can 
keep  the  wheels  moving  against  resistance  on  account  of 
the  fact  that  some  one  has  done  work  upon  it  in  winding 
it  up.  Energy  possessed  by  a  body  because  of  its  position  or 
form  is  called  potential  energy.  The  potential  energy  of  a 
body  is  measured  by  the  work  that  was  done  upon  it  to 
bring  it  into  the  condition  by  virtue  of  which  it  possesses 
that  energy.  Hence  a  body  whose  mass  is  100  pounds 
which  has  been  lifted  a  distance  of  5  feet  has  500  foot- 
pounds of  potential  energy,  i.e.  it  is  able  to  do  500  foot- 
pounds of  work  because  of  its  elevated  position. 


56  A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

61.  Kinetic  Energy.  —  When  a  lifted.  weight  is  allowed 
to  fall,  it  does  work  upon  the  object  that  it  strikes.     At 
the  instant  of  striking  it  possesses  energy  because  it  is  in 
motion.     Moreover,  any  moving  body  is  able  to  do  work 
by  virtue  of  its  motion.      The  energy  possessed  by  a  body 
because  of  its  motion  is  called  kinetic  energy.     The  kinetic 
energy  of  a  body  is  measured  by  the  amount  of  work  done 
upon  it  to  put  it  in  motion. 

62.  Kinetic  Energy  Computed.  —  The  kinetic  energy  of  a 
moving  body  is  measured  by  one  half  its  mass  multiplied  by 
the  square  of  its  velocity.     This  may  be  shown  in  the  fol- 
lowing manner  :     Let  a  body  whose  mass  is  m  grams  be 
acted  upon  by  a  force  of  /  dynes  which  will  give  it  an 
acceleration   of    a   cm.  /sec.2*      From   (2),   §   36,  /=  ma. 
Again,  since   the   force   produces   uniformly  accelerated 
motion   in  the  given  mass,  at  the  end  of   t  seconds,  as 
shown   by   (3),    §  25,  the   body   will    have   been    moved 
through    a    distance    d  =  J  at2.        Now    the    velocity    v 
acquired  by  the  mass  in  t  seconds  as  shown  by  (2),  §  25,  is 

v  =  at  cm./sec.  ;  whence  t2  =  —  .     Substituting  this  value 

for  t2  in  the  equation  for  distance,  we  obtain  d  =  —  centi- 

meters. 

In  order  to  compute  the  work  done  by  the  force  /, 
we  have  only  to  multiply  the  force  by  the  distance  d 
through  which  it  acts  (§  55).  Thus 


Work  =fd  =  max-  =         ergs. 
£a         A 

Since  the  work  done  in  producing  the  motion  is  the 
measure  of  the  kinetic  energy  of  the  mass  m  (§  59), 

Kinetic  Energy  =  \  mv2  ergs.  (2) 


WORK   AND  ENERGY  57 

Let  the  mass,  velocity,  and  acceleration  be  given  in  the  units  of 
the  F.  P.  S.  system.  Then  the  product  ma  will  give  the  force  in 

poundals  (§  36),  and  the  quantity  — ,  the   distance  through   which 

the  force  acts,  in  feet.     Therefore  the  product  of  force  and  distance 
will  give  the  work  in  foot-poundals.     Hence,  in  the  F.  P.  S.  system 

Kinetic  Energy  =  \  mv2  foot-poundals. 

It  should  be  remembered  that  the  formula  for  kinetic 
energy  deduced  above  gives  the  result  in  the  absolute 
system  only,  i.e.  in  ergs  or  foot-poundals.  These  can  be 
readily  reduced  to  kilogram-meters  and  foot-pounds  by 
the  help  of  the  numerical  relations  given  in  §  56. 

EXAMPLE.  —  Calculate  the  kinetic  energy  of  a  10-gram  bullet 
whose  velocity  is  40,000  cm./sec. 

SOLUTION.  —  Using  equation  (2),  we  have  for  the  kinetic  energy 
of  the  bullet  \  x  10  x  40,000  x  40,000,  or  8,000,000,000  ergs.  By  re- 
ferring to  §  54,  we  observe  that  1  kg-m.  equals  98,000,000  ergs. 
Therefore,  the  reduction  from  ergs  to  kilogram-meters  gives  for  the 
kinetic  energy  of  the  body  8163  kg-m. 

EXERCISES 

1.  Calculate  the   potential   energy  given  to  a  mass  of  25  g.  by 
lifting  it  through  a  vertical  height  of  10  m.     Express  the  result  in 
kilogram-meters. 

2.  A  ball  moving  with  a  velocity  of  3500  cm./sec.  has  a  mass  of 
250  g.     Find  its  kinetic  energy  in  ergs.     How  much  work  must  a 
boy  do  in  order  to  stop  it  ? 

3.  Compare  the  kinetic  energy  of  the  ball  in  Exer.  2  with  that  of 
a  mass  of  25,000  g.  whose  velocity  is  350  cm./sec. 

4.  What  is  the  kinetic  energy  of   a  5-gram   bullet  just  as  it  is 
leaving  the  muzzle  of  a  gun  with  a  velocity  of  30,000  cm./sec.? 

5.  To  what  height  would  the  bullet  in  Exer.  4  have  to  be  taken 
in  order  to  have  an  equal  amount  of  potential  energy  ? 

SUGGESTION.  —  First  find  the  force  required  to  lift  the  bullet  in 
dynes;  then  apply  equation  (1),  §  55. 

6.  Compute  the  kinetic  energy  of  a  5-pound  mass  moving  with  a 
velocity  of  25  ft.  per  second.     Express  the  result  in  foot-pounds. 

SUGGESTION.  —  First  obtain  the  result  in  foot-poundals ;  then  re- 
duce to  foot-pounds  by  the  help  of  §  56. 


58  A  HIGH   SCHOOL  COURSE   IN  PHYSICS 

7.  A  constant  force  of  200  dynes  acts  upon  a  mass  of  5  g.     Cal- 
culate (1)  the  acceleration,  (2)  the  velocity  produced  in  3  seconds, 
and  (3)  the  kinetic  energy.     What  is  the  distance  through  which 
the  force  acts  during  the  3  seconds  ? 

8.  In  order  to  move  a  load  up  a  hill  250  ft.  long,  a  horse  exerts 
a  constant  pull  of   125  Ib.     How  much  work  is  done  ?     If  the  load 
weighs  900  Ib.,  to  what  height  would  an  equivalent  amount  of  work 
lift  it? 

9.  A  stone  whose  mass  is  50  kg.  is  placed  on  the  top  of  a  chimney 
30  m.  in  height.     Calculate  the  amount  of  work  that  must  be  per- 
formed in  kilogram-meters  and  foot-pounds. 

10.  Compute  the  amount  of  work  done  per  minute  by  a  pumping 
engine  that  forces  100,000  gal.  of  water  into  a  reservoir  120  ft.  high 
every  10  hr.     Assume  the  density  of  water  to  be  62.5  Ib.  per  cubic 
foot. 

11.  If  a  rifle   ball  whose   mass  is  8  g.  has  a  velocity  of  35,000 
cm. /sec.,  how  far  will  it  penetrate  a  block  of  wood  that  offers  a  uni- 
form resistance  of  100,000  g. 

SUGGESTION. — Let  x  be  the  depth  of  penetration  in  centimeters, 
and  place  the  work  done  by  the  ball  expressed  in  ergs  equal  to  the 
kinetic  energy. 

12.  The  elevation  of  a  tank  containing  25,000  gal.  of  water  is 
75  ft.     Find  the  potential  energy  of  the  water. 

4.    TRANSITIONS   OF   ENERGY 

63.  Transference  and  Transformation  of  Energy.  —  No 
processes  in  nature  are  of  more  common  occurrence  than 
transferences  of  energy  from  one  body  of  matter  to  an- 
other and  transformations  of  energy  from  one  form  into 
another.  For  example,  if  a  body  is  allowed  to  fall  freely, 
the  potential  energy  that  it  possesses  while  elevated  is  grad- 
ually transformed  into  kinetic  energy  which  resides  in  the 
body  until  its  motion  is  checked.  If,  however,  the  body 
should  fall  upon  a  spring  properly  placed,  the  spring 
would  be  compressed  and  thus  possess  potential  energy  at 
the  expense  of  the  kinetic  energy  of  the  falling  mass. 
Hence  energy  is  transferred  from  the  falling  body  to  the 
spring.  Whenever  one  ~body  does  work  upon  another,  energy 


WORK   AND   ENERGY 


59 


is  transferred  from  the  body  that  does  the  work  to  the  one 
upon  which  the  work  is  done. 

Let  the  elevated  mass  M,  Fig.  27,  be  suspended  by  a  cord  wound 
around  an  axle  A  to  which  is  at- 
tached a  heavy  wheel  W.  It  is 
plain  that  the  downward  pull  of 
M  upon  the  cord  will  cause  the 
wheel  to  turn.  Thus,  as  M  falls 
and  loses  potential  energy,  it 
does  work  upon  the  wheel  in  pro- 
ducing motion  and  thus  impart- 
ing kinetic  energy. 

When  the  cord  is  fully  un- 
wound, the  action  will  not  cease  ; 
but  the  kinetic  energy  of  the 
wheel,  by  winding  up  the  cord 


FIG.  27.  —  Transformation  of  the  Po- 
tential Energy  of  the  Raised  Mass 
M  into  Kinetic  Energy  in  the 
Wheel  W. 


around  the  axle  on  the  opposite 
side,  will  enable  it  to  lift  M.  In 
this  manner  the  wheel  performs 
work  upon  M,  losing  its  kinetic  energy  and  imparting  potential  en- 
ergy to  the  mass  lifted.  If  no  energy  were  lost  in  overcoming  fric- 
tion, the  kinetic  energy  imparted  to  the  wheel  in  the  former  case 
would  be  completely  restored  to  the  mass  M  in  the  latter. 

Changes  in  energy  occur  in  a  large  number  of  common 
processes,  such  as  winding  a  clock  or  a  watch,  shooting 
an  arrow  from  a  bow,  running  a  sewing  machine,  turning 
a  grindstone,  running  mills  by  water  power,  etc. 

64.  Conservation  of  Energy.  —  Although  energy  is  pass- 
ing continually  through  transformations  and  is  being 
transferred  from  one  body  to  another  around  us  on  every 
hand,  no  one  has  ever  been  able  to  prove  that  even  the 
smallest  portion  can  be  created  or  destroyed.  The  infer- 
ence is,  therefore,  that  the  same  quantity  of  energy  is  present 
in  the  universe  to-day  as  existed  ages  ago  ;  i.e.  that  the  quan- 
tity of  energy  present  in  the  universe  remains  constant. 
This  is  known  as  the  Law  of  the  Conservation  of  Energy. 


60  A.  HIGH   SCHOOL  COURSE   IN  PHYSICS 

The  principal  aim  of  Physics  is  to  trace  the  various  trans- 
formations and  transferences  of  energy  that  accompany 
natural  phenomena.  At  this  point  in  the  study  many  of 
these  changes  will  seem  obscure  because  all  the  forms  in 
which  energy  may  exist  have  as  yet  not  been  considered. 
For  example,  we  may  inquire  what  becomes  of  the  kinetic 
energy  of  a  spinning  top  as  it  slowly  comes  to  rest.  As 
we  pursue  the  study  further,  we  find  that  where  there  is 
motion  in  opposition  to  friction,  as  in  this  case,  heat  is 
produced.  But  heat  is  one  of  the  forms  that  energy  may 
take.  Hence  the  kinetic  energy  of  the  top  will  appear 
somewhere  in  the  form  of  heat. 

65.  Matter  and  Energy.  —  The  intimate  relation  be- 
tween matter  and  energy  is  becoming  more  and  more 
apparent.  Matter  is  obviously  a  carrier,  or  vehicle,  of  en- 
ergy. We  become  acquainted  with  matter  only  through 
natural  phenomena.  In  each  phenomenon  there  is  in- 
volved some  change  in  energy,  and  it  is  in  the  transforma- 
tions and  transferences  of  energy  that  our  senses  are 
affected.  It  is  upon  these  processes  that  we  base  our 
entire  knowledge  of  the  material  world. 

EXERCISES 

1.  In  driving  a  well  a  heavy  weight  is  elevated  by  a  horse  and 
then  allowed  to  fall  upon  the  end  of  a  vertical  pipe,  thus  forcing  it 
into  the  ground.     Trace  the  energy  changes  taking   place   in  the 
process. 

2.  Trace  the  transferences  and  transformations  of  energy  in  the 
process  of   driving   a   nail;    of  planing   a  board;    of    shooting   an 
arrow ;    of  throwing  a  stone ;    of   winding  a  clock ;    of  running  a 
sewing  machine;   of  beating  an  egg. 

3.  Account  for  the  energy  of  the  water  above  a  dam  in  a  river. 
What  becomes  of  this  energy  ? 

4.  In  what  form  is  a  supply  of  energy  taken  on  board  an  ocean 
steamer  ?     In  what  form  is  energy  supplied  to  a  locomotive  ?  to  an 
automobile?  to  a  horse?  to  a  man? 


WORK  AND   ENERGY  61 

SUMMARY 

1.  The  term  work  is  used  to  express  the  process  of  pro- 
ducing motion.     Work  involves  both  force  and  motion  in 
the  direction  of  the  force,  and  is  measured  by  the  product 
of  the  force  employed  multiplied  by  the  distance  through 
which  it   acts.     The  equation  of  work  is   Work  =/X  d 
(§  55). 

2.  The  erg,  or  dyne- centimeter,  is  the  C.  G.  S.   unit  of 
work  and  of  energy  and  is  the  work  done  by  a  force  of  one 
dyne  acting  through  a  distance  of  one  centimeter.     The 
foot-poundal  is  the  English  absolute  unit  of  work  and  en- 
ergy.    The  kilogram-meter  and  foot-pound  are  the  gravita- 
tional units  in  common  use  (§  56). 

3.  The  activity,  or  power,  of  an  agent  is  the  rate  at 
which  it  can  do  work.     The  activity  of  an  agent  is  said  to 
be  one  horse  power  when  it  can  perform  work  at  the  rate 
of  550  foot-pounds  (or  746  x  10r  ergs)  per  second  (§§57 
and  58). 

4.  The  energy  of  a  body  is  its  capacity  for  doing  work, 
or  its  ability  to  do  work  (§  59). 

5.  Potential  energy  is  the  energy  possessed  by  a  body 
because  of  its  position  or  form  (§  60). 

6.  Kinetic  energy  is  the  energy  possessed  by  a  body  by 
virtue  of  its  motion.     The  equation  of  kinetic  energy  is 
K.E.  =  l  mv*  (§§  61  and  62). 

7.  When  one  body  does  work  upon  another,  energy  is 
transferred  from  the  body  that  does  the  work  to  the  one 
upon  which  the  work  is  done  (§  63). 

8.  Energy  cannot  be  created  or  destroyed.     The  quan- 
tity present  in  the  universe  remains  constant.     This  is 
known  as  the  Law  of  the  Conservation  of  Energy  (§  64). 


CHAPTER  V 
GRAVITATION 

1.    LAWS   OF  GRAVITATION  AND  WEIGHT 

66.  Universal  Gravitation.  —  Ancient  astronomical  ob- 
servations revealed  the  fact  that  the  planets  move  through 
space  in  curvilinear  paths.     Later  and  more  refined  obser- 
vations led  to  the  discovery  that  the  sun  is  a  center  about 
which  they  revolve  in  slightly  elliptical  orbits.     Further- 
more, it  is  universally  known  that  several  of  the  planets 
have   satellites   which   revolve    about   them,    correspond- 
ing to  the  moon  which  moves  in  an  orbit  encircling  the 
earth.     Late  in  the  seventeenth  century  Sir  Isaac  Newton 
originated  the  theory  that  is  now  known  as  the  Law  of 
Universal  Gravitation,  in  order  to  account  for  the  motion 
of  heavenly  bodies  in  nearly  circular  orbits  instead  of 
straight  lines. 

67.  Newton's  Law  of  Universal  Gravitation.  —  This  law 
may  be  stated  as  follows  : 

Every  body  in  the  universe  attracts  every  other  body  with 
a  force  which  is  directly  proportional  to  the  product  of  the 
attracting  masses  and  inversely  proportional  to  the  square  of 
the  distance  between  their  centers  of  mass  (§  70). 

According  to  this  law  a  book  and  a  marble,  or  two 
bodies  of  any  other  kind  of  matter,  attract  each  other. 
Between  ordinary  masses  this  force  remains  unnoticed  by 
us  in  everyday  life  because  it  is  so  minute ;  in  fact,  it 
would  require  the  most  refined  test  to  detect  it.  But 
since  the  attraction  is  proportional  to  the  product  of  the 
masses,  in  the  case  of  two  heavy  bodies  —  the  moon  and 

62 


GRAVITATION  63 

the  earth,  for  example  —  the  force  is  enormous.  Even 
between  the  earth  and  a  marble  or  a  book  the  force  is 
quite  perceptible.  When  the  earth  is  one  of  the  acting 
masses,  the  attraction  is  called  the  force  of  gravity,  and 
when  expressed  in  the  proper  units  of  measure,  this  attrac- 
tion is  called  the  weight  of  the  marble,  book,  etc.  Weight, 
therefore,  partakes  of  the  nature  of  a  force  and  is  quite 
distinct  from  mass  (§  10).  Hence,  when  we  say  in 
ordinary  language,  for  instance,  that  the  intensity  of  a 
certain  force  is  10  pounds,  we  mean  that  it  is  equal  to 
that  force  with  which  the  earth  attracts  a  mass  of  10 
pounds.  Again,  since  weight  is  a  force,  it  may  be  ex- 
pressed in  any  of  the  units  of  force  (§§  36  and  37),  i.e.  in 
dynes,  poundals,  etc. 

68.  Weight.  —  Since  for  a  given   locality  the  mass  of 
the  earth,  as  well  as  the  distance  from  the  center,  is  con- 
stant, the  weight  of  a  body  is  strictly  proportional  to  its  mass. 
Again,    since    the    earth   is    not    spherical    but   flattened 
slightly  at  the  poles,  the  same  mass  at  different  places 
will  not  possess  the  same  weight.     On  moving  north  or 
south  from  the  'equator  the  radius  of  the  earth  decreases 
slightly,  which  causes  the  mass  to  come  somewhat  nearer 
the  earth's  center  and  thus  increases  the  value  of   the 
force  of  attraction. 

69.  Law  of  Weight. — The  law  of    universal  gra vita- 
ion  applied   to  bodies  outside  the  earth's  surface  is  as 
follows  : 

The  weight  of  a  body  above  the  earth's  surface  is  inversely 
proportional  to  the  square  of  its  distance  from  the  center  of 
the  earth. 

If  the  radius  of  the  earth  is  assumed  to  be  4000 
miles,  the  weight  of  a  one-pound  mass  4000  miles 
above  the  surface,  which  is  8000  miles  from  the  cen- 
ter, would  be  only  one  fourth  of  a  pound.  This  result 


64  A  HIGH   SCHOOL  COURSE  IN  PHYSICS 

is  obtained  by  applying  the  law  of  weight  as  follows : 

x  :  1  pound  :  :  40002  :  80002  ; 
whence  x  =  J  pound. 

Since  the  distance  from  the  earth's  center  is  less  at 
Chicago  than  at  the  equator,  a  mass  of  1  pound  weighs 
about  g-^g-  of  a  pound  more  at  the  former  place  than  at 
the  latter.  For  small  differences  of  latitude,  however, 
the  difference  in  weight  is  so  small  that  it  is  of  little 
importance.  In  consequence  of  the  earth's  rotation  the 
weight  of  bodies  at  the  equator  is  diminished  ^9  (§  53) 
as  the  result  of  the  centrifugal  reaction  against  the  force 
of  gravity.  In  other  latitudes  this  diminution  is  less. 

It  is  of  interest  to  consider  what  the  effect  would  be  upon  the 
weight  of  a  given  mass  if  it  were  to  be  taken  to  some  point  below 

the  surface  of  the  earth.     Let  it  be  im- 
agined that  the  circle  in  Fig.  28  repre- 
sents a  cross  section  through  the  center 
of  the  earth,  and  that  P  is  the  location 
of   the  body  to  be  weighed.      All  that 
part  of   the   earth    represented    by  the 
shaded  portion  of  the  circle  above  the 
plane  AB  will  exert  a  resultant  attrac- 
tion upward,  while  that  represented  by 
the  unshaded  part  has  a  resultant  acting 
FIG.  28.  —  A  Mass  at  P  is  At-     toward  the  center  O.     Since  these  forces 
tracted  Upward  as  well  as     oppose  each  other,  the  weight  of  the  body 
Downward.  .,,    ,.     .    .  ,          .,  ,       „_, 

will  dimmish  as  it  approaches  the  center 

O.  When  the  body  reaches  the  center,  the  attractions  due  to  the 
different  portions  of  the  earth  will  be  equal  in  all  directions,  and  the 
resultant  of  all  will  be  zero.  Therefore  the  body  will  weigh  nothing 
at  the  center  of  the  earth. 

EXERCISES 

1.  If  the  mass  of   the  earth  were  doubled  without  any  change 
in  its  shape  or  size,  how  would  a  person's  weight  be  affected? 

2.  Which  is  a  definite  quantity,  a  gram  of  matter  or  a  gram  of 
force  (i.e.  a  gram-weight)  ? 


GRAVITATION 


65 


3.  How  much  will  the  potential  energy  of  a  mass  of  2000  fb. 
elevated  100  ft.  at  Chicago  differ  from  that  of  an  equal  mass  raised 
100ft.  at  the  equator? 

4.  A  certain  mass  is  weighed  on  a  dynamometer  (§  11)  at  New 
York.     Will  the  instrument  indicate  a  greater  or  a  less  weight  when 
the  same  mass  is  weighed  at  the  equator  ? 

5.  If  two  masses  are  in  equilibrium  when  placed  in  the  pans  of 
a  beam  balance  at  the  equator,  will  they  still  be  in  equilibrium  when 
tested  in  the  same  manner  at  San  Francisco? 

6.  How  far  above  the  earth's  surface  would  a  body  weigh  one  half 
as  much  as  at  the  surf  ace  ?  Ans.     1656.8  mi. 

7.  What  would  a  100-pound  body  weigh  at  a  distance  of  200  ini. 
above  the  earth's  surface  ? 

8.  An  aeronaut  ascends  5  mi.  in  a  balloon.     If  his  weight  at  the 
surface  is  150  lb.,  what  will  it  be  at  that  height  ? 


2.   EQUILIBRIUM   AND    STABILITY 

70.  Center  of  Gravity.  —  The  weight  of  a  body  is  the 
resultant  of  the  weights  of  the  individual  particles  of 
which  it  is  composed.  Since  these 
innumerable  forces  are  all  directed 
toward  the  center  of  the  earth,  which 
is  4000  miles  away,  they  form  a  sys- 
tem of  essentially  parallel  forces 
whose  resultant  CA,  Fig.  29,  is  equal 
to  their  sum  (§  46).  The  point  of 
application  O  is  called  the  center  of 
gravity  of  the  body.  Since  the  posi- 
tion of  this  point  depends  upon  the 
distribution  of  matter  in  the  body,  it 
is  also  called  the  center  of  mass.  In  many  problems  it  is 
convenient  to  consider  the  body  as  though  all  its  mass 
were  located  at  this  point.  If  a  flat  piece  of  cardboard  of 
any  shape  is  balanced  on  the  point  of  a  pin,  the  center  of 
gravity  is  located  at  the  point  of  contact  and  midway 
between  the  two  surfaces. 
6 


FIG.  29. —  Weight  is  the 
Resultant  of  Innumer- 
able Small  Forces. 


66 


A  HIGH   SCHOOL  COURSE  IN  PHYSICS 


Let  a  flat  piece  of  cardboard  be  pierced  at  any  point,  as  A,  Fig.  30, 
and  hung  loosely  on  a  small  nail  or  pin.  The  cardboard  will  turn 
until  the  center  of  gravity  falls  as  low  as  pos- 
sible. In  this  condition  a  vertical  line  through 
A  will  pass  through  the  center  of  gravity. 
This  line  is  easily  found  by  hanging  a  plumb 
line  from  the  axis  in  front  of  the  cardboard. 
If  a  second  point  of  support,  as  £,  be  taken 
and  a  vertical  line  determined  as  before,  the 
center  of  gravity  C  will  lie  at  the  point  of  in- 
tersection of  the  two  lines. 
FIG.  30.  —  Locating  the 

Center  of  Gravity.  ^      Equilibrium  Of  Bodies.  —  A  body 

is  said  to  be  in  equilibrium  when  a  vertical  line  through 
the  center  of  gravity  passes  through  a  point  of  support. 
A  common  case  of  equi- 
librium is  that  of  a  chair 
or  table.  In  such  in- 
stances a  vertical  line' 
through  the  center  of 
gravity  passes  through 
the  area  of  the  base  in- 
cluded within  the  lines 
joining  the  feet.  Four 


FIG.  31.  — Stable,  Unstable,  and  Neutral 
Equilibrium  Illustrated. 


typical  cases  are  represented 
in  Fig.  31.  Pyramid  A  hangs  from  its  apex,  B  stands 
upon  its  base,  and  C  rests  with  its  apex  at  the  point  of 
support.  Obviously  A  and  B  tend  to  remain  indefinitely 
in  the  positions  shown,  but  0  will  overturn  with  the 
slightest  disturbance.  If  A  or  B  should  be  tilted,  the 
center  of  gravity  would  be  lifted,  necessitating  the 
expenditure  of  energy  upon  the  body.  A  and  B  are  said 
to  be  in  stable  equilibrium.  On  the  other  hand,  a  disturb- 
ance of  0  lowers  the  center  of  gravity  and  thus  lessens 
the  potential  energy  of  the  body.  When  a  body  is  in 
this  condition,  it  is  said  to  be  in  unstable  equilibrium. 
A  third  condition  is  represented  by  a  sphere  of  uniform 


GRAVITATION  67 

density  resting  upon  a  smooth  horizontal  plane.  If  the 
sphere  be  rolled  along  the  plane,  its  center  of  gravity  will 
be  neither  raised  nor  lowered.  It  is  said  to  be  in  neutral 
equilibrium.  A  body  arranged  to  turn  upon  an  axis 
through  its  center  of  gravity  is  also  in  a  condition  of 
neutral  equilibrium. 

72.  Stability  of  Bodies.  —  When  a  body  is  in  stable 
equilibrium,  work  must  be  performed  upon  it  in  order  to 
cause  it  to  overturn.  This  amount  of  work  will  depend 
upon  the  weight  of  the  body  and  the  distance  through 
which  its  center  of  gravity  is  lifted,  and  is  measured  by 
their  product  (§  55).  The  amount  of  work  required  to 
overturn  a  body  is  a  measure  of  its  stability. 

EXAMPLE.  —  Find  the  stability  of  a  box  4  ft.  square  and  2  ft.  high 
and  weighing  500  Ib. 

SOLUTION.  —  Referring  to  Fig.  32,  it  is  plain  that  the  center  of 
gravity  C  must  be  moved  to  the 
point  A  while  the  box  is  being  over- 
turned. The  height  through  which 
C  is  raised  is  BA,  equal  to  OC  — 
OB.  Now  OC  is  the  hypotenuse 
of  the  right  triangle  OBC  whose 
sides  are  1  ft.  and  2  ft.  respectively. 
Hence  OC  equals  V5",  or  2.24  ft. 

Therefore,  AB  is  1.24  ft.,  and  the __ 

work  done  1.24  x  500,  or  620  foot-  FlG.  32.  — Center  of  Gravity  is  Lifted 
pounds.  Through  the  Height  BA. 

It  is  clear  that  of  two  bodies  having  the  same  weight, 
the  more  stable  one  is  that  whose  center  of  gravity  has  to 
be  lifted  through  the  larger  vertical  distance  when  we 
overturn  it.  This  will  depend  on  the  size  and  shape 
of  the  base  on  which  it  rests  and  on  the  height  of  the 
center  of  gravity  above  the  base.  Figure  33  shows  a  brick 
in  three  possible  positions.  The  center  of  gravity  C 
moves  through  an  arc  having  the  lower  right-hand  corner 


68 


A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


of  the  brick  as  its  center  when  the  body  is  overturned 
about  this  point.     It  will  be  seen  at  once  that  the  greatest 


i 

~^7 

\ 

\ 

/ 

1 

i 

i 
i 

\ 

\ 

1 
1 
1 

a    1 

\ 

»^                                                      \ 

\ 

^v                                                     \ 

I 

i 
i 

/ 

>          >n  n 

/   \ 

\ 
\ 
\ 
\ 

i 

,'c 

c  /                                         /                  \ 

\ 

/ 

i 

^\ 

/                                               \ 

^-~> 

(/)  (2)  (3) 

FIG.  33.  —  The  Overturning  of  a  Brick  about  Different  Edges. 

stability  is  possessed  by  the  brick  when  lying  on  its  largest 
base ;  first,  because  the  base  is  largest,  and  second,  because 
the  center  of  gravity  is  in  the  lowest  possible  position. 
In  each  case  c'a  is  the  vertical  distance  through  which  the 
center  of  gravity  would  have  to  be  lifted  by  the  overturn- 
ing agent. 

Note  the  various  methods  employed  to  give  the  proper 
stability  to  objects  in  everyday  use,  as  lamps,  clocks,  ink- 
stands, chairs,  pitchers,  vases,  etc. 

EXERCISES 

1.  How  would  you  place  a  cone  on  a  horizontal  table  in  positions 
representing  the  three  conditions  of  equilibrium  ? 

2.  Arrange  two  knives  in  a  piece  of  wood  as  shown  in  Fig.  34  and 

support  the  point  on  the  finger.  Why  is  the 
system  in  stable  equilibrium?  Where  is  the 
center  of  gravity  of  the  system  ? 

3.  Why  is  it  difficult  to  walk  on  stilts? 

4.  Explain  the  difficulty  experienced  in  try- 
ing to  balance  an  upright  rod  upon  the  end  of 
the  finger. 

5.  Why  does   not  the   Leaning   Tower    of 
Pisa  fall?    See  Fig.  36. 

6.  Explain  the  difficulty  experienced  in  trying  to  balance  a  meter 
stick  on  one  end  upon  a  level  table. 


FIG.  34.  — A  System  in 
Stable  Equilibrium. 


GRAVITATION 


69 


7.  The  oil  can  B  shown  in  Fig.  35  is  loaded  with  lead  at  the  bot- 
tom.    Explain  how  this  can  will  right  itself  while  one  of  the  common 
form  A  remains  overturned. 

8.  Which  of  two  bodies  hav-  A 
ing  equal  weights  possesses  the 

greater  stability,  a  pyramid  or  a 
rectangular  box  having  the  same 
base  and  height  as  the  pyramid?  FIG.  35.  —  Oil  Cans. 

SUGGESTION.  —  The  center  of  gravity  of  a  pyramid  is  located  at 
one  third  of  the  distance  from  the  base  to  the  apex. 

9.  Calculate  the  stability  in  foot-pounds  of  a  4-pound  brick  placed 
in  three  different  positions  on  a  horizontal  table.    Assume  the  dimen- 
sions to  be  2  x  4  x  8  in.     See  example  in  §  72. 

3.  THE   FALL  OF  UNSUPPORTED  BODIES 

73.  Falling  Bodies.  —  Before  the  time  of  the  Italian 
mathematician  and  physicist,  Galileo  Galilei,1  little 
was  known  concerning  the  way  in  which 
bodies  fall  when  unsupported.  It  is  a 
well-known  fact  that  if  we  drop  a  coin 
and  a  piece  of  paper  or  a  feather  at  the 
same  instant,  the  coin  will  reach  the  floor 
first.  Galileo  rightly  inferred  that  the 
difference  was  due  to  the  resistance  of- 
fered by  the  air.  Nevertheless,  in  order 
to  place  on  an  experimental  basis  his 
conclusion  that  all  falling  bodies  tend 
to  have  the  same  acceleration,  he  dropped 
bodies  of  different  kinds  from  the  top 
of  the  Leaning  Tower  of  Pisa  (Fig. 
36)  in  the  presence  of  many  learned  men  of  the  time. 
These  experiments  demonstrated  that  all  bodies  tend  to  fall 
from  a  given  height  in  practically  equal  times.  Furthermore, 
it  was  readily  shown  that  light  materials,  as  paper,  for  ex- 
ample, fall  in  less  time  when  compressed  than  when  spread 
1  See  portrait  facing  p.  70. 


FIG.  36.  —  Leaning 
Tower  of  Pisa, 
Italy. 


70  A  HIGH   SCHOOL   COURSE   IN   PHYSICS 

out.  Later,  after  the  invention  of  the  air  pump,  Galileo's 
inference  was  verified  by  allowing  a  light  and  a  heavy 
body  to  fall  in  a  vacuum.  For  this  purpose 
the  "  guinea  and  feather  "  tube  (Fig.  37)  is 
commonly  used.  On  inverting  the  tube 
after  the  air  has  been  exhausted,  we  find 
that  the  feather  falls  as  rapidly  as  the  coin. 
But  when  the  air  is  again  admitted,  the 
feather  flutters  slowly  along  far  behind  the 
rapidly  falling  coin. 

74.    Uniform  Acceleration  of  Falling  Bodies. 
—  Since   the  attraction  existing  between  a 
body  and  the  earth  is  constant,  it  follows  that 
a  body  falling  freely,  i.e.   without  encoun- 
tering resistance,  will  have  uniformly  accel- 
erated motion  (§§  24  and  36).     Again,  since 
FIG.  37.— Bodies  a^  bodies  fall  the  same  distance  in  a  given 
Fall  Alike  in  time  when  unimpeded,  the  acceleration  will 
be  the  same  for  all  bodies.     This  acceleration 
is  called  the  acceleration  due  to  gravity,  and  is  designated 
by  the  letter  g. 

75.  The  Acceleration  Due  to  Gravity.  —  The  acceleration 
of  freely  falling  bodies  varies  according  to  the  laws  of 
weight  given  in  §  69.  Since  the  distance  to  the  center  of 
the  earth  decreases  as  one  travels  from  the  equator  toward 
the  poles,  the  acceleration  due  to  gravity  becomes  greater. 
In  latitude  38°  N.  g  is  980  cm./sec.2,  and  in  latitude  50°  N. 
981  cm./sec.2.  The  value  of  g  also  decreases  slightly  with 
the  elevation  above  sea-level.  (Why  ?)  In  the  latitude 
of  New  York,  40.73°  N.,  a  freely  falling  body  gains  in  ve- 
locity at  the  rate  of  about  980  centimeters,  or  32.16  feet, 
per  second  during  each  second  of  its  motion. 

When  bodies  are  thrown  upward,  the  acceleration  is  negative ;  i.e. 
the  velocity  decreases  at  the  rate  of  980  centimeters  per  second 


GALILEO    GALILEI    (1564-1642) 

The  first  successful  experimental  investigations  relating  to  falling 
bodies  and  the  pendulum  must  be  attributed  to  Galileo.  For  nearly 
twenty  centuries  the  science  of  Mechanics  had  remained  undeveloped. 
Aristotle  had  announced  that  the  rate  at  which  a  body  falls  depends 
upon  its  weight,  but  Galileo  was  the  first  to  disprove  it  by  experi- 
ment. This  he  did  by  dropping  light  and  heavy  bodies  from  the 
leaning  tower  of  Pisa,  Italy,  his  native  town.  A  one-pound  ball  and 
a  one-hundred-pound  shot,  which  were  allowed  to  fall  at  the  same 
time,  were  observed  by  a  multitude  of  witnesses  to  strike  the  ground 
together.  Hence  the  rate  of  fall  was  shown  to  be  independent  of 
mass. 

At  another  time,  while  observing  the  swinging  of  a  huge  lamp 
in  the  cathedral,  Galileo  was  astonished  to  find  that  the  oscillations 
were  made  in  equal  periods  of  time  no  matter  what  the  amplitude. 
He  proceeded  to  test  the  correctness  of  this  principle  by  timing  the 
vibrations  with  his  own  pulse.  Later  in  life  he  applied  the  pendu- 
lum in  the  construction  of  an  astronomical  clock. 

Galileo  was  the  first  to  construct  a  thermometer  and  the  first  to 
apply  the  telescope,  which  he  greatly  improved,  to  astronomical 
observations.  He  discovered  that  the  Milky  Way  consists  of  innu- 
merable stars;  he  first  observed  the  satellites  of  Jupiter,  the  rings  of 
Saturn,  and  the  moving  spots  on  the  sun. 

Galileo  was  made  professor  of  mathematics  in  the  University  of 
Pisa  in  1589  and  filled  a  similar  position  at  Padua  from  1592  until 
1610.  He  died  in  the  year  1642,  the  year  of  Newton's  birth. 


GRAVITATION 


71 


during  each  second  of  its  upward  motion.  Hence,  a  body  thrown 
upward  with  a  velocity  of  2940  centimeters  per  second  will  continue 
to  rise  3  seconds,  when  it  will  stop  and  return  to  earth  in  the  next  3 
seconds.  However,  on  account  of  the  hindrance  of  the  air,  bodies 
moving  with  great  velocity  deviate  considerably  from  the  laws  govern- 
ing unimpeded  bodies. 

76.  Laws  of  Freely  Falling  Bodies.  —  Since  the  motion 
of  an  unimpeded  body  while  falling  is  uniformly  acceler- 
ated, the  equations  of  §  25  may  be  applied  by  simply  sub- 
stituting for  a  the  acceleration  due  to  gravity  g.  These 
equations  may  be  written  as 
follows  : 

v  =  gt,  (1) 

and  d  =  I  gt2.  (2) 

From  equations  (l)  and  (2) 
other  useful  formulae  may  be 
deduced.  From  (l)  we  find  that 


dfor  1  s«c  = 


AtB,     V= 


dfor  2  sec=4  x  ^  g  -"  ^ 


At  C,     v^2g  -  - 


AtD,     v=3g 


7  x/29 


t2  =  —  •     Substituting  this  value 
for  t2  in  equation  (2),  we  obtain 


Solving  equation  (3)    for   v, 
we  have 

v  =  V2~gd.  (4) 

77.  Distances  and  Velocities 
Represented.  —  The  distances 
passed  over  and  the  velocities 
acquired  by  a  freely  falling  body 
are  represented  graphically  in 
Fig.  38.  A  vertical  line  is 
drawn  on  which  a  convenient  distance  AS  is  measured  off 
to  represent  1  x  \g  (about  16  feet),  the  distance  the 
body  falls  during  the  first  second.  The  distance  AC  is 


FIG.  38.  —  Motion  of  a  Falling 
Body  Represented. 


72 


A  HIGH   SCHOOL  COURSE  IN  PHYSICS 


c' 


£' 


made  four  times  the  distance  AB ;  AD,  nine  times  AB ; 
AE,  sixteen  times  AB,  etc.,  to  represent  the  distances 
fallen  in  one,  two,  three,  and  four  seconds  respectively. 
The  heavy  arrows  are  drawn  to  represent  the  velocity 
at  the  end  of  each  second.  The  length  of  the  first  is 
g  units  (representing  about  32  feet  per  second),  the 
second  2#,  the  third  3#,  etc. 

78.  Bodies  Thrown  Horizontally.  —  It  is  a  well-known 
fact  that  a  body  projected  in  any  direction  except  up  or 
down  follows  a  curved  path.  An  interesting  case  of  this 

kind  is  the  projection  of  a  body 
horizontally  from  some  elevated 
position,  as  A,  Fig.  39.  The 
motion  of  the  body  will  be  the 
resultant  of  two  component  mo- 
tions, the  one  vertically  down- 
ward due  to  gravity,  and  the 
other  in  a  horizontal  direction 
due  to  the  projectile  force. 
Since  there  is  no  horizontal 
force  acting  on  the  body  after  it 
leaves  the  point  A,  the  horizon- 
tal component  of  the  motion 
will  bi 


F.o.39.-MotionofaBodyPro- 

jected  Horizontally  from  the    will   move    (horizontally)  over 
'omt  A.  equal  distances  in  equal  intervals 

of  time.  Let  AB' ,  B1 Cf,  C'D',  etc.,  represent  these  hori- 
zontal distances  for  successive  seconds.  The  distances  AB, 
BO,  CD,  etc.,  are  drawn  in  the  manner  described  in  §  77. 
Under  the  combined  action  of  its  initial  horizontal  velocity 
and  the  force  of  gravity  the  body  will  pass  through  the 
point  P1  at  the  end  of  the  first  second,  P2  at  the  end  of  the 
second,  P3  at  the  end  of  the  third,  etc.  Thus  the  body 
follows  the  curved  path 


GRAVITATION  73 

The  path  of  a  stone  thrown  over  a  tree,  for  example,  is  a  case  in 
which  the  initial  motion  of  the  body  is  not  horizontal.  The  motion 
may  be  divided  into  two  parts  :  first,  the  rise  of  the  stone  to  the  highest 
point  reached ;  and,  second,  the  fall  of  the  body  back  to  the  earth. 
The  latter  half  of  the  motion  is  precisely  the  case  described  above. 
The  time  during  which  the  stone  is  rising  is  practically  equal  to  that 
of  its  fall.  The  time  required  for  the  return  of  the  stone  to  the  earth 
is  the  same  as  that  of  a  body  falling  vertically.  Likewise  the  time 
occupied  by  the  stone  in  rising  is  equal  to  that  required  by  a  body 
thrown  vertically  upward  to  attain  the  same  height.  The  shape  of 
the  path  taken  by  the  stone  depends  on  its  initial  speed  and  the  direc- 
tion in  which  it  is  thrown.  The  study  of  the  motion  of  projected 
bodies,  as  bullets,  cannon  balls,  shells,  etc.,  forms  an  important  part  of 
military  and  naval  instruction. 


EXERCISES 

1.  A  body  falls  freely  from  a  certain  height   and  reaches  the 
ground  in  5  seconds.    What  velocity  is  acquired?    From  what  height 
must  it  fall? 

2.  How  long  does  it  take  a  body  to  fall  100  ft.  ?   200  ft.  ? 

3.  A  mass  of  50  g.  falls  for  3  seconds  from  a  state  of  rest.     Calcu- 
late its  kinetic  energy.     (See  §  62.) 

4.  A  mass  of  50  Ib.  falls  from  an  elevation  of  20  ft.     Calculate  its 
kinetic  energy  in  foot-pounds  at  the  time  it  reaches  the  ground.     Com- 
pare the  kinetic  energy  with  the  potential  energy  of  the  body  before 
falling. 

5.  How  far  must  a  body  fall  in  order  to   acquire  a  velocity  of 
500  ft.  per  second? 

6.  A  book  falls  from  a  table  3  ft.  in  height.     Find  the  velocity  of 
the  book  when  it  reaches  the  floor. 

7.  A  stone  is  dropped  from  a  train  whose  velocity  is  30  mi.  per 
hour.     Show  by  a  diagram  the  path  traced  by  the  stone.     (See  §  78.) 

8.  Find  the  kinetic  energy  of  a  10-gram  mass  after  it  has  fallen 
from  rest  a  distance  of  1960  cm.,  assuming  y  to  be  980  cm./sec.2. 

9.  The  velocity  of  a  body  falling  freely  from  rest  was  200  ft.  per 
second.     From  what  height  did  it  fall? 

10.  Compare  the  velocity  of  a  body  after  falling  64.32  ft.  with 
that  of  a  train  running  30  mi.  per  hour. 

11.  A  bullet  is  fired  vertically  upward  from  a  gun  with  a  velocity 


74  A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

of  25,000  cm./sec.  Disregarding  the  resistance  of  the  air,  how  many 
seconds  will  the  bullet  continue  to  rise?  How  high  will  it  rise? 

12.  If  the  bullet  in  Exer.  11  encountered  no  resistance  due  to  the 
air,  how  many  seconds  would  pass  before  it  returned  to  earth  ? 

13.  The  weight  of  a  pile-driver  is  lifted  10  ft.  and  allowed  to  fall. 
With  how  much  greater  velocity  will  it  strike  if  lifted  20  ft.  ?    With 
how  much  greater  energy  ? 

14.  A  stone  thrown  over  a  tree  reaches  the  earth  in  3  seconds. 
What  is  the  height  of  the  tree? 

15.  A  boy  fires  a  rifle  ball  vertically  upwards  and  hears  it  fall  upon 
the  ground  in  20  seconds.     How  high  does  it  rise?     What  was  its 
initial  velocity? 

4.  THE  PENDULUM 

79.    The    Simple    Pendulum.  —  A    heavy   particle    sus- 
pended from  a  fixed  point  by  a  weightless  thread  of  constant 

length  is  an  ideal  simple  pendu- 
lum. If,  however,  we  suspend 
a  small  metal  ball  A,  Fig.  40, 
by  a  thin  flexible  thread  or  wire 
from  a  fixed  point  0,  it  fulfills 
the  ideal  conditions  almost  per- 
fectly. The  distance  OA  from 
the  point  of  suspension  to  the 
center  of  the  ball  is  the  length 
of  the  pendulum.  The  arc  AC, 
or  the  angle  AOC,  which  rep  re- 
ef^ &\c  sents  the  displacement  of  the 
9 ball  from  the  position  of  equi- 
librium, is  the  amplitude  of  vi- 

FIQ  40.  —  The  Simple  Pendulum.     ,7  .,         .        . 

oration.     A  vibration  is  one  to- 

and-fro  swing,  sometimes  called  a  complete  or  double  vibra- 
tion. A  single  vibration  is  the  motion  of  the  ball  from  C 
to  C1,  or  one  half  of  a  complete  vibration.  The  period  of 
a  single  vibration  is  the  time  consumed  by  the  pendulum 
in  moving  from  C  to  C1 . 


GRAVITATION 


75 


80.  Pendular  Motion  Due  to  Gravity.  —  When  a  pen- 
dulum is  in  the  position  of  rest,  the  weight  of  the  ball  rep- 
resented by  the  line  A  W,  Fig.  41,  is  balanced  by  the  equal 
and  oppositely  directed  ten- 
sion AT  in  the  cord  OA. 
Now  let  the  ball  be  drawn 
aside  to  the  position  C  and 
released.  The  weight  of  the 
ball,  which  always  acts  ver- 
tically downward  and  is 
represented  by  the  line  6YP, 
can  be  resolved  into  two 
components.  One  of  these 
components  is  represented 
by  the  line  CE  and  serves  FIG.  41. -Gravity  Causes  a  Pendulum 
solely  to  produce  tension  in  to  Vibrate, 

the  cord  00.  It  is  plain  that  this  component  has  no  effect 
on  the  motion  of  the  ball.  The  other  component  CB  acts 
upon  the  ball  along  the  tangent  to  the  arc  of  vibration  at 
the  point  0.  The  effect  of  this  component  is  to  give  the 
ball  accelerated  motion  along  the  arc.  After  the  ball  passes 
the  point  A,  it  is  clear  that  the  component  along  the  tangent 
tends  to  retard  the  motion  and  finally  succeeds  in  stopping 
the  ball  at  the  point  <7',  after  which  it  returns  the  ball 
again  to  A. 

If  the  line  CD  is  drawn  perpendicular  to  OA,  the  triangles 
CDO  and  CBP  are  similar.  Why?  We  may  therefore  write  the 
proportion 

CB  :  CP  : :  CD  :  CO  (5) 

This  proportion  may  also  be  written  as  follows : 

Force  CB  =  displacement  CD  x  weight  of  ball  CP .  (6) 

length  CO 

Since  the  weight  of  the  ball  CP  and  the  length  of  the  pendulum 
CO  remain  constant  during  a  vibration,  equation  (6)  shows  that  the 
effective  force  CB  is  proportional  to  the  displacement  CD.  Hence 


76 


A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


the  acceleration  of  the  ball  is  not  the  same  at  all  points  in  the  arc 
CA,  but  varies  directly  as  the  displacement. 

81.    Transformations  of  Energy  in  the  Pendulum.  —  A 

pendulum  is  first  set  in  motion  by  displacing  the  ball,  or 
pendulum  bob.  This  process  requires  the  performance  of 
work  which  elevates  the  ball  through  the  height  AD,  Fig. 
41  (measured  vertically),  and  stores  potential  energy  in  it. 
From  equation  (l),  §55,  it  is  clear  that  the  amount  of 
potential  energy  given  the  ball  is  measured  by  the  product 
of  its  weight  and  the  height  AD.  As  the  ball  moves  toward 
A,  velocity  is  acquired,  and  the  potential  energy  is  gradu- 
ally changed  into  kinetic.  At  A  the  energy  is  all  kinetic. 
After  passing  the  point  A  the  ball  rises  to  C',  while  the 
kinetic  energy  is  transformed  back  into  potential.  At  O1 
the  energy  is  all  potential  again.  Thus  recurrent  trans- 
formations of  energy  take  place,  which  would  occur  with- 
out loss  if  it  were  not  for  the  resistance  offered  by  the 
air  as  well  as  by  friction  at  the  point  of 
suspension. 

82.  Laws  of  the  Simple  Pendulum.  — 
The  first  three  laws  of  the  simple  pendu- 
lum may  be  deduced  from  the  results 
obtained  from  the  following  experiments  : 

Suspend  four  balls  as  shown  in  Fig.  42.  Let 
A,  B,  and  C  be  of  metal,  and  D  of  wood  or  wax. 
Make  the  lengths  of  A,  B,  and  C  as  1:4:9;  e.g. 
20,  80,  and  180  cm.  Also  let  Z>  be  made  precisely 
of  the  same  length  as  C.  Now  if  C  and  D  be  set 
swinging  through  the  same  amplitude,  it  will  be 
readily  observed  that  the  period  of  vibration  of  D 
is  the  same  as  that  of  C. 

Again,  let  C  and  D  be  set  in  vibration  through 
different  amplitudes.    If  neither  amplitude  is  large, 
it  will  be  seen  that  the  period  of  one  is  still  the 
same  as  that  of  the  other. 

Finally,  let  A,  B,  and  C  be  put  in  motion  successively  and  the 


A  BCD 


FIG.  42. —  Pendu- 
lums of  Differ- 
ent Lengths  and 
Masses. 


GRAVITATION  77 

single  vibrations  of  each  counted  for  one  minute.  If  the  period  of 
vibration  of  each  pendulum  be  computed  from  the  number  of  vibra- 
tions per  minute,  it  will  be  found  that  the  three  periods  are  as  1  :  2  :  3, 
i.e.  as  VI:  VI:  VQ. 

The  laws  governing  the  vibration  of  simple  pendulums, 
therefore,  may  be  stated  thus  : 

(1)  The  period  of  vibration  is  independent  of  the  material, 
or  mass,  of  the  ball. 

(2)  When  the  amplitude  of  vibration  is  small,  the  period 
of  vibration  is  independent  of  the  amplitude;  i.e.  the  vibra- 
tions are  made  in  equal   times.     This  is  called   the  Law 
of  Isochronism  (pronounced  i  sok'ro  nism).     If  the  ampli- 
tude exceeds  5°  or  6°,  the  period  of  vibration  will  gradually 
diminish  as  the  arc  becomes  smaller. 

(3)  The  period  of  vibration  is  directly  proportional  to 
the  square  root  of  the   length  of  the  pendulum.     This  is 
called  the  Law  of  Length.     This  law  may  be  represented 
thus  :    ^  :  t2  :  :  Vl1  :  VI2,  where  ^  and  ^  refer  to  the  period 
and  length  of  one  pendulum,  and  t2  and  Z2,  to  the  period 
and  length  of  the  other.     If,  therefore,  any  three  of  the 
terms  are  given,  the  fourth  may  be  computed. 

Since  a  pendulum  is  dependent  upon  the  force  of  gravity, 
as  shown  in  §  80,  its  period  of  vibration  is  found  to  depend 
upon  the  value  of  g.  Hence  : 

(4)  The  period  of  vibration  is  inversely  proportional  to 
the  square  root  of  the  acceleration  due  to  gravity. 

83.  The  Pendulum  Equation.  —  The  relation  between 
the  period  and  length  of  a  simple  pendulum  and  the 
acceleration  due  to  gravity  is  given  by  the  equation 


where  t  is  the  period  of  a  single  vibration,  I  the  length  of 
the  pendulum  measured  in  centimeters  (or  feet),  and  g 


78  A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

the  acceleration  due  to  gravity  measured  in  centimeters 
per  second  per  second  (or  feet  per  second  per  second). 
The  value  of  TT  is  3.1416,  the  ratio  of  the  circumference 
of  a  circle  to  the  diameter. 

This  equation  is  of  great  assistance  (1)  in  calculating 
the  period  of  any  simple  pendulum  of  known  length,  (2) 
in  determining  the  length  of  a  pendulum  that  vibrates  in 
any  given  period  of  time,  and  (3)  in  finding  the  value  of 
g  at  any  place  where  its  magnitude  is  unknown.    (See  §  87. ) 
84.    The  Seconds  Pendulum.  —  A  pendulum  of  which  the 
period  of  a   single  vibration   is  one   second   is   a   seconds 
pendulum.     As  shown  by  equation  (7),  the  length  of  a 
simple  pendulum  £,  when  the   period   t  is 
one  second,  will  depend  on  the  acceleration 
due  to  gravity  at  the  place  chosen.     At  all 
places  where  tfce  value  of  g  is  980  cm. /sec.2, 
the  length  I  of  a  simple  pendulum  that  beats 
seconds  is  99.3  cm. ;  where  g  is  981  cm. /sec.2, 
I  is  99.4  cm. 

85.    The   Compound   Pendulum.  —  When 
the  conditions  defining  the  ideal  simple  pen- 
dulum (§  79)  are  not  sufficiently  fulfilled, 
the  body  is  called  a  compound  or  physical 
pendulum.       For   experimental   purposes  a 
meter  bar  may  be  siispended  on  a  smooth 
wire  nail  which  pierces  it  at  right  angles 
close  to  one  end.     The  bar  may  be  hung  to 
.  —  A  Com-  swing  freely  between  the  prongs  of  a  large 
pound  Pendu-  tuning  fork,  as  shown  in  Fig.  43.     Let  a 

lum    and     its      .        _&  '  .  .  .  .          _         '     .         .         .  , 

Equivalent  simple  pendulum  OA  be  placed  by  the  side 

fumPle  Pendu"  °f  tne  suspended  bar  so  that  their  points  of 

suspension  lie  in  the  same  horizontal  plane. 

Set  both  pendulums  in  vibration  and  adjust  the  length  of 

the  simple  one  until  they  vibrate  in  the  same  period  of 


GRAVITATION  79 

time.  It  will  be  observed  at  once  that  the  simple  pendu- 
lum must  be  made  several  inches  shorter  than  the  other. 
Only  those  points  in  the  compound  pendulum  very  near 
the  point  O  swing  in  their  natural  period  ;  particles  below 
0  tend  to  swing  slower,  and  those  above,  faster,  than  the 
simple  pendulum.  O  is  called  the  center  of  oscillation  of 
the  compound  pendulum.  In  this  case  the  point  c  is  lo- 
cated two  thirds  of  the  length  of  the  bar  from  the  point 
of  suspension.  The  simple  pendulum  is  thus  seen  to  be  a 
special  case  of  the  compound  one  in  which  the  entire  mass 
is  concentrated  near  the  center  of  oscillation. 

The  compound  pendulum  in  Fig.  43  may  be  set  swinging 
by  being  struck  a  sharp  blow  at  the  point  6y,  and  the  axis 
will  not  be  disturbed.  For  this  reason  O  is  called  the  center 
of  percussion.  Thus,  for  example,  when  a  ball  is  batted, 
the  bat  should  be  so  handled  that  the  ball  will  strike  its 
center  of  percussion.  This  will  prevent  the  jarring  of  the 
hands  and  the  breaking  of  the  bat. 

86.  The  Compound  Pendulum  Reversible.  — If  the  meter 
bar  shown  in  Fig.  43  be  suspended  by  piercing  it  at  the 
center  of  oscillation  0  and  swinging  it  about  this  point, 
the  period  of  vibration  will  be  the  same  as  before.     In 
other  words,  the  point  of  suspension  B  and  the    center 
of    oscillation     C    are    interchangeable.     This    property 
of  a  compound  pendulum,  which  was  discovered  by  the 
famous     Dutch     physicist     Huyghens     (1629-1695),    is 
known  as  its  reversibility. 

87.  Utility  of  the  Pendulum.  —  The  value  of  the  pendu- 
lum in  the  measurement  of  time  is  due  to  the  isochronism 
of  its  vibrations.     Although  Galileo  was  the  first  to  ob- 
serve this  property  of  the  pendulum  and  the  first  to  make 
a  drawing  of  a  pendulum  clock,  Huyghens  was  the  first  to 
use  a  pendulum  in  controlling  the  motion  of  the  wheels  of 
a  timepiece.     This  he  accomplished  in  1656. 


80 


A  HIGH   SCHOOL  COURSE  IN   PHYSICS 


The  motion  of  a  clock  is  maintained  by  lifted  weights  or  by  the 
elasticity  of  springs.  The  office  of  the  wheelwork  is  to  move  the 
hands  over  the  dial  and  to  keep  the  pendulum  from 
being  brought  to  rest  by  friction.  The  latter  is 
effected  by  means  of  the  escapement  shown  in 
Fig.  44.  The  wheel  R  is  turned  by  mechanism 
not  shown  in  the  figure.  When  the  pendulum 
swings  to  the  right,  motion  is  communicated  to  the 
curved  piece  MN  through  the  parts  A,  B,  and  O ; 
and  M  is  lifted.  The  wheel  is  thus  released ;  but, 
on  turning,  strikes  at  N.  While  the  pendulum 
moves  to  the  left,  the  slight  pressure  of  the  cog 
against  N  causes  A  to  deliver  a  minute  force  to  the 
pendulum.  As  N"  rises,  the  wheel  is  again  re- 
leased, but  is  again  detained  a,t'M.  Thus  one  cog 
is  allowed  to  pass  for  each  complete  vibration  of  the 
pendulum.  Every  "tick"  of  the  clock  is  caused 
by  the  wheel  R  being  stopped  either  at  M  or  N. 
If  the  wheel  is  allowed  to  turn  too  fast,  the  clock 
gains  time.  This  defect  is  corrected  by  lowering 
the  bob.  If  the  clock  loses  time,  the  bob  is  raised. 

The   pendulum  offers  the  most  precise 
method  for  measuring  the  acceleration  of 
FIQ  a  gravity.    By  carefully  determining  experi- 

ment and   Pen-  mentally  the  period  t  and  the  length  I  of  a 
duiumofadock.  pendulum,  the  value  of  g  can  be  easily  cal- 
culated by  the  help  of  equation  (7),  §  83. 

EXERCISES 

1.  By  the  help  of  equation  (7),  §  83,  find  the  period  of  a  pendulum 
80  cm.  long,  when  g  equals  980  cm. /sec.2. 

2.  Calculate  the  length  of  a  simple  pendulum  that  beats  half 
seconds  (i.e.  t  equals  |  sec.)  at  a  place  where  the  acceleration  is  981 
cm./sec.2. 

3.  The  pendulum  of  a  clock  has  a  period  of  a  quarter  second. 
Find  its  length  if  g  is  980  cm./sec.2. 

4.  How  long  is  a  simple  pendulum  that  makes  65  single  vibrations 
per  minute? 

SUGGESTION.  —  First  compute  the  value  of  t. 


GRAVITATION  81 

5.  What  is  the  value  of  g  where  a  simple  pendulum  99.2  cm.  long 
makes  60  single  vibrations  per  minute? 

6.  An  Arctic  explorer  finds  that  the  length  of  the  seconds  pendu- 
lum at  a  certain  place  is  99.6  cm.     What  is  the  value  of  g  at  this  place  ? 

7.  A  simple  pendulum  is  to  make  45  single  vibrations  per  minute. 
If  g  is  980  cm./sec.2,  what  must  be  its  length? 

8.  It  is  found  at  a  certain  place  that  a  simple  pendulum  90  cm. 
long  makes  64  single  vibrations  per  minute.     Find  the  value  of  g  at 
this  place. 

9.  A  pendulum  whose  bob  weighs  100  g.  is  drawn  aside  until  the 
distance  AD,  Fig.  41,  is  4  cm.     How  much  energy  is  stored  in  the 
bob  ?     How  much  work  was  done  upon  it? 

SUMMARY 

1.  Newton's  Law   of  Universal  G-ravitation  states  that 
every  body  in  the  universe  attracts  every  other  body  with 
a  force  that  is  directly  proportional  to  the  product  of  the 
attracting  masses  and  inversely  proportional  to  the  square 
of  the  distance  between  their  centers  of  mass  (§  67). 

2.  The  attraction  of  the  earth  for  other  bodies  is  called 
the  force  of  gravity.     The  weight  of  a  body  is  the  measure 
of  this  force  (§  67). 

3.  The  weight  of  a  body  is  proportional  to  its  mass  (§  68). 

4.  The  weight  of  a  body  above  the  earth's  surface  is  in- 
versely proportional  to  the  square  of  its  distance  from  the 
center  of  the  earth.     On  account  of  the  spheroidal  form  of 
the  earth,  a  body  at  the  equator  weighs  slightly  less  than 
a  body  of  the  same  mass  at  some  other  point  on  the  earth's 
surface  (§  69). 

5.  The  weight  of  a  body  is  the  resultant  of  the  weights 
of  the  individual  particles  that  compose  it.     The  point  of 
application  of  this  resultant  is  the  center  of  gravity  of  the 
body  (§  70). 

6.  A  body  is  in  equilibrium  when  a  vertical  line  drawn 
through  its  center  of  gravity  passes  through  a  point  of 

7 


82  A  HIGH  SCHOOL  COURSE  IN   PHYSICS 

support,  or  within  the  area  included  between  the  extreme 
points  of  support.  The  three  kinds  of  equilibrium  are 
stable,  unstable,  and  neutral  (§  71). 

7.  The  stability  of  a  body  is  measured  by  the  work  that 
must  be  performed  in  order  to  overturn  it  (§  72). 

8.  All  freely  falling  bodies  descend  from"  the  same 
height  in  equal  times.     Such  bodies  have  uniformly  ac- 
celerated motion.     The  acceleration  due  to  gravity  is  about 
980  cm./sec.2,  or  32.16  ft./sec.2  (§§  73-75). 

9.  The  equations  of  freely  falling  bodies  are  (1)  v  =  gt, 

(2)  d  =  \gt\  (3)  d=  ^  and  (4)  v  =  V2^d  (§  76). 

10.  When  bodies  are  thrown,  the  horizontal  component 
of  the  motion  is  uniform,  while  the  vertical  component  is 
uniformly  accelerated  (§  78). 

11.  The  swinging  of  a  pendulum  is  due  to  the  force  of 
gravity  (§  80). 

12.  The  period  of  vibration  o^a  pendulum  is  independ- 
ent of  the  mass  of  the  bob  and\the  amplitude  of  vibra- 
tion when  the  arc  is  small,  and  is  tiirectly  proportional  to 
the  square  root  of  the  length  andNinversely  proportional 
to  the  square  root  of  the  acceleration  due  to  gravity.    The 

equation  of  the  pendulum  is  t  =  TT  -y^  (§  82). 

13.  A  compound  pendulum  may  be  conceived  as  being 
made  up  of  simple  pendulums  of  different  lengths.     The 
period  of  vibration  depends  upon  the  position  of  the  center 
of  oscillation.     The  center  of  oscillation  and  point  of  sus- 
pension   are   interchangeable.     The  center  of  percussion 
and  the  center  of  oscillation  coincide  (§§  85  and  86). 


CHAPTER  VI 

MACHINES 
1.    GENERAL   LAW   AND   PURPOSE    OF   MACHINES 

88.  Simple  Machines.  —  The  transference  of  energy  from 
a  body  capable  of  doing  work  to  another  upon  which  the 
work  is  to  be  done  is  often  accomplished  more  advanta- 
geously by  the  use  of  a  simple  machine  than  in  any  other 
way.     Indeed,  it  is  often  impossible  for  an  agent  to  do 
the  required  work  without  the  aid  of  a  machine.     For 
example,  a  man  wishes  to  load  a  barrel  of  lime  into  a 
wagon,  but  finds  that  he  is  unable  to  lift  it ;   with  the 
aid  of  an  inclined  plane  of  suitable  length,  however,  the 
barrel  is  easily  rolled  into  the  wagon.     In  other  cases  use 
is  made  of  the  pulley,  lever,  wheel  and  axle,  screw,  and  wedge, 
which  with  the  inclined  plane  form  the  six  simple  machines. 

89.  The    Principle    of    Work.  —  The 
general  law  of  machines  is  illustrated  by 
the  following  experiments : 

1.  Let  a  cord  be  passed  over  a  pulley,  as  shown 
in  Fig.  45.  Let  the  pull  of  a  dynamometer  be 
used  to  counteract  the  weight  of  the  body  W. 
It  is  obvious  in  this  case  that  the  amount  of  force 
F  registered  on  the  dynamometer  must  be  equal 
to  the  weight  W.  The  experiment  will  show  FIG.  45. — The  Effort 
that  this  is  the  case.  Furthermore,  if  force  F  F  .  Equals  the 
moves  downward  1  foot,  W  will  be  elevated  Moves  Througlian 
through  an  equal  distance.  Equal  Distance. 

If,  now,  we  designate  the  distance  through  which  the 
acting  force   (or  effort)  F  moves  by  the  letter  d  and  the 

83 


84 


A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


distance  the  weight  Wis  lifted  by  d1  ',  the  work  put  into  the 
machine  by  the  acting  agent  is  F  x  d,  and  the  work  done 
by  the  machine  is  W  x  d'  .  It  is  plain  that  the  experiment 
shows  that  - 


2.  Let  the  pulley  be  now  attached  to  the  weight  W,  Fig.  46,  and 
let  one  end  of  the  cord  be  fastened  to  some  stationary  object  at  A .  If 
W  is  made  1000  grams,  for  example,  it  will  be 
found  that  the  upward  effort  registered  by  the 
dynamometer  will  be  500  grams.  For  any  value 
of  W,  F  will  be  one  half  as  great.  However,  when 
F  moves  a  distance  of  1  meter,  for  example,  W  is 
elevated  only  one  half  a  meter. 

In  this  experiment  W=  2  F,  and  d'  ==  \  d. 
Therefore,  we  may  write  as  before 

Fxd  =  Wxd'.  (l) 

The  relation  shown  by  this  equation  is 
one  of  the  most  important  laws  of   me- 
Fio.   46.— TheEf-  chanics  and  is  known  as  the  Principle  of 
One  Half  ofUthe  Work.     The   principle   may  be  stated  as 

Weight   W,   but    follows: 
Moves  Twice  as 

Far-  The  work  done  by  an  agent  upon  a  ma- 

chine is  equal  to  the  work  accomplished  by  the  machine ;  or, 
the  effort  F  multiplied  by  the  distance  through  which  it  ads 
equals  the  resistance  W  that  is  overcome  by  the  machine  mul- 
tiplied by  the  distance  it  is  moved. 

This  law  may  be  applied  to  any  machine,  no  matter  how 
simple  or  complicated  it  may  be,  provided  the  friction  of 
the  moving  parts  can  be  disregarded. 

EXAMPLE.  —  An  agent  capable  of  doing  work  exerts  a  force  of 
50  Ib.  upon  a  machine.  If  a  weight  of  250  Ib.  is  lifted  8ft.  by  the 
machine,  through  what  distance  must  the  applied  force  act? 

SOLUTION.  —  The  work  done  by  the  machine  is  250  x  8,  or  2000 
foot-pounds.  Hence,  the  force  applied  by  the  agent  must  act  through 
the  distance  2000  -4-  50,  or  40  ft. 


MACHINES  85 

It  is  clear  from  this  example  that  the  gain  in  force  is 
accomplished  at  the  expense  of  distance,  since  the  effort 
must  move  five  times  as  far  as  the  resistance.  Its  speed 
also  is  five  times  as  great.  On  the  other  hand,  a  machine 
may  be  made  to  increase  the  distance  as  well  as  the  speed 
at  the  expense  of  force.  Such  is  the  case  in  many  practi- 
cal applications  of  the  simple  machines. 

90.  Mechanical  Advantage  of   a  Machine.  —  It  is   fre- 
quently desirable  to  know  the  multiplication  of  force  that 
is  brought  about  by  the  use  of  a  machine ;   or,  in  other 
words,  the  ratio  of  the  resistance  W  to  the  effort  F.     This 
ratio  is  called  the  mechanical  advantage  of  the  machine. 

From  equation  (l)  we  may  write 

W:F::d:d'.  (2) 

Hence,  the  mechanical  advantage  of  a  machine  is  the  ratio 
of  the  distance  through  which  the  effort  moves  to  the  distance 
through  which  the  resistance  is  moved  by  the  machine. 

For  example,  the  mechanical  advantage  of  the  pulley  as 
used  in  the  first  experiment  of  the  preceding  section  is  1, 
in  the  second  experiment  2,  and  in  the  example  given 
on  page  84  it  is  5. 

2.    THE    PRINCIPLE    OF   THE    PULLEY 

91.  The   Pulley.  —  The   pulley  consists   of   a   grooved 
wheel,  called  a  sheave,  turning  easily  in  a   block   that 
admits  of  being  readily  attached  to  objects.     When  the 
block  containing  the  sheave  is  attached  to  some  stationary 
object,  the  pulley  is  said  to  be  fixed;   when   the  block 
moves  with  the  resistance,  the  pulley  is  movable.     A  fixed 
pulley  is  shown  in  (1),  Fig.  47,  and  a  single  movable 
pulley  in  (2). 

Let  experiments  be  made  with  pulleys  arranged  as  shown  in  Figs. 
47  and  48.  In  each  case  ascertain  by  means  of  a  dynamometer  01 


86 


A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


weights  the  force  required 
at  F  to  balance  a  weight 
applied  at  W.  The  effect  of 
friction  may  largely  be 
avoided  by  taking  the  mean 
of  the  forces  applied  at  F, 
first  when  W  is  slowly  raised 
and  then  when  it  is  slowly 
lowered. 

When  a  single  fixed 
pulley  is  used,  W=  F. 
When  a  single  movable 


FIG.  47.—  (1)  A  Fixed  Pulley. 
(2)  A  Single  Movable  Pul- 
ley. (3)  A  System  of  One 
Fixed  and  One  Movable 
Pulley. 

pulley  is  used,  the  re- 
sistance is  applied  to 
the  movable  block,  as 
shown  in  (2),  Fig.  47. 
A  study  of  the  figure 
will  show  that  W  is 
balanced  by  two  equal 
parallel  forces,  each  of 
which  is  equal  to  F. 
ReiiceW^ZF.  The 
mechanical  advantage 
is  therefore  2.  In  (3) 

FIG.  48.  — (1)   One  Movable   and  Two  Fixed    also      the      mechanical 

Pulleys.    (2)  Two  Movable  and  Two  Fixed  advantage  is  2,  and  the 

Pulleys.      (3)    Two    Movable    and    Three         .  ? 

Fixed  Pulleys.  only  gam  secured   by 


MACHINES 


87 


the  use  of  the  fixed  pulley  is  one  of  direction,  i.e.  the  effort 
may  now  act  downward  instead  of  upward  as  in  (2). 

In  (1),  Fig.  48,  one  end  of  the  rope  is  attached  to  the 
movable  block,  so  that  W  is  supported  by  three  upward 
parallel  forces,  each  of  which  is  equal  to  F.  Hence 
W=  3  F.  A  similar  consideration  of  (2)  will  show  that 
W=  4  F,  and  of  (3),  that  W=  5  F. 

It  is  obvious  from  the  cases  already  considered  that 
whenever  a  continuous  cord  is  used  in  the  pulley  system, 
the  mechanical  advantage  is  equal  to  the  number  of  paral- 
lel forces  acting  against  the  resistance  W.  If  n  is  the 
number  of  the  parts  of  the  rope  supporting  the  mova- 
ble block,  then  n  is  the  number  of  these  parallel  and  equal 
forces  of  which  Wis  the  sum.  Therefore 

W  =  nF.  (3) 

92.  Principle  of  Work  and  the  Pulley  System. — Equa- 
tion (3)  can  be  derived  by  applying  the  general  law  of 
work  stated  in  §  89.  If  there  are  n  portions  of  the  rope 
supporting  the  movable  pulley,  and  W  is  lifted  1  foot, 
for  example,  each  portion  of  the  cord  must  be  shortened 
that  amount.  Consequently  the  effort 
F  must  move  n  feet.  By  the  principle 
of  work  the  product  of  the  effort  and 
the  distance  through  which  it  acts 
equals  the  resistance  multiplied  by  the 
distance  through  which  it  is  moved  ;  or, 


EXERCISES 

1.  Diagram  a  set  of  pulleys   by  means  of 
which  an  effort  of  100  Ib.  can  support  a  load  of 
500  Ib. 

2.  What  is  the  mechanical  advantage  of  the 
system  of  pulleys  shown  in  Fig.  49  V     Find  the 
effort  required  to  balance  a  weight  of  1200  Ib. 


FIG.  49.  — A  Tackle. 


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A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


3.  Each  of  two  pulley  blocks  contains  two  sheaves.     Show  by  a 
diagram  how  to  arrange  these  into  a  system  that  will  enable  an  effort 
of  75  Kg.  to  move  a  resistance  of  300  Kg. 

4.  Show  by  a  diagram  the  best  arrangement  of  two  blocks,  one 
containing  two  sheaves,  the  other  containing  one.     Ascertain  the 
mechanical  advantage. 

5.  In  each  case  shown  in  Fig.  48  let  the  effort  be  applied  to  the 
movable  block  and  the  resistance  W  to  the  end  of  the  rope  in  place  of 
F.     If  the  effort  F  moves  1  ft.,  how  far  will  W  be  moved  ?    State  the 
advantage  secured  in  each  instance.     NOTE.  —  This  plan  is  frequently 
employed  in  the  operation  of  passenger  elevators  in  tall  buildings. 

6.  In  a  pulley  system  consisting  of  a  continuous  cord  attached  at 
one  end  to  a  movable  block  containing  one  sheave,  the  rope  passes 
through  a  fixed  block  having  two  sheaves.     Find  the  effort  required 
to  support  a  block  of  marble  weighing  a  ton. 

3.  THE  PRINCIPLE  OF  THE  LEVER 

93.    The  Lever.  —  Of  all  the  simple  machines  the  lever 
is  the  most  common.     It  is  of  frequent  occurrence  in  the 

structure  of  the  skeleton  of 
man  and  animals,  as  well  as 
in  many  mechanical  appli- 
ances. In  its  simplest  form 
the  lever  is  a  rigid  bar,  as 
AB,  Fig.  50,  arranged  to  turn  about  a  fixed  point  0  called 
the  fulcrum.  The  effort  is  applied  at  the  point  J.,  and  the 
resistance  at  B. 

Balance  a  meter  stick  upon  a  long  wire  nail,  piercing  it  a  little  above 
the  center,  as  shown  in  Fig. 
51.  By  means  of  a  small 
cord  or  thread,  suspend  a 
weight  of  150  grains  at  B,  15 
centimeters  from  the  ful- 
crum 0.  Now  place  a 
weight  of  50  grams  upon 
the  opposite  side  of  the 
fulcrum  and  move  it  along 
the  bar  until  it  just  balances 


FIG.  50.  — The  Lever. 


so  g 


FIG.  51.— The  Moment  of  the  Effect  F 
equals  the  Moment  of  the  Resistance 
W. 


MACHINES  89 

the  weight  at  B.  Call  this  point  A.  It  will  now  be  found  that  the 
distance  A  0  is  45  centimeters.  From  this  it  is  plain  that  the  force  F 
multiplied  by  the  arm  AO  is  equal  to  the  weight  W  multiplied  by  the 
arm  BO.  The  experiment  may  be  extended  by  using  other  forces  and 
distances,  but  in  every  case  it  will  be  found  that  the  product  F  x  AO 
equals  the  product  W  x  BO. 

The  product  of  the  applied  force  F  multiplied  by  its 
distance  AO  from  the  fulcrum  is  called  the  moment  of 
that  force  (§  45).  Likewise,  the  product  of  the  resist- 
ance W  multiplied  by  its  distance  BO  from  the  fulcrum 
is  the  moment  of  the  resisting  force.  Thus  the  fact  shown 
by  the  experiment  may  be  stated  as  follows  : 

The  moment  of  the  effort  is  equal  to  the  moment  of  the 
resistance. 

The  distances  AO  and  BO  are  called  the  arms  of  the 
lever.  Representing  these  distances  by  I  and  V  respec- 
tively, the  law  of  the  lever  is  represented  by  the  equa- 
tion 

F  x  1=  W  x  1'.  (4) 

From  this  equation  it  is  clear  that  the  mechanical  ad- 

W  I 

vantage  —  (§  90)  is  represented  by  the  ratio  -•     Hence,  a 

.  I 

lever  will  support  a  weight  or  other  resistance   —  times  as 

great  as  the  effort. 

94.  Classes  of  Levers.  —  Levers  are  usually  divided  into 
three  classes  (Fig.  52)  depending  upon  the  relative  location 
of  the  fulcrum,  the  effort,  and  the  resistance. 

(1)  In  levers  of  the  first  class  the  fulcrum  is  between 
the  effort  F  and  the  resistance  W\  as  in  the  crowbar,  beam 
balance,  scissors,  steelyard,  pliers,  wire  cutters,  etc. 

(2)  Levers  of  the   second   class   are   distinguished   by 
the  fact  that  the  resistance  W  is  between  the  fulcrum  0 
and  the  effort  F-,  as  in  the  nutcracker,  wheelbarrow,  etc. 


90 


A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


(8)  Levers  of  the  third  class  are  distinguished  by  the 
fact  that  the  effort  F  is  between  the  fulcrum  0  and  the 
resistance  W\  as  in  the  fire  tongs,  sheep  shears,  sewing- 
machine  treadle,  etc. 

F\ 


1st  Class. 


2d  Class. 
FIG.  52. —  Levers  of  the  Three  Classes. 


If  the  forces  acting  on  the  lever  are  not  parallel,  it  is 
spoken  of  as  a  bent  lever.  A  hammer  used  in  pulling  a 
nail  (Fig.  53)  is  a  lever  of  this  kind. 
Bent  levers  are  frequently  found  in 
complicated  machines ;  as  in  farming 
implements,  metal- working  machin- 
ery, clocks,  etc. 

In  levers  of  all  classes  the  arms  are 
measured  from  the  fulcrum  0  on  lines 
perpendicular  to  the  lines  which  rep- 
resent the  direction  of  the  forces. 
The  law  stated  in  the  preceding  sec- 
tion holds  for  all  cases. 

95.  Extension  of  the  Principle  of 
Moments.  —  The  relation  of  the  mo- 
ments of  the  forces  acting  on  the  arms  of  a  lever  as  ex- 
pressed in  §  93  may  be  extended  to  cases  in  which  any 
number  of  forces  act  upon  the  bar,  provided  they  pro- 
duce equilibrium.  The  case  may  be  illustrated  by  experi- 
ment as  follows: 


mum 

FIG.  53.  — The  Hammer 
used  as  a  Bent  Lever. 


MACHINES  91 

Let  weights  be  placed  on  the  balanced  meter  stick  used  in  §  93  pre- 
cisely as  shown  in  Fig.  54.     The  forces  will  be  found  to  balance. 

Since  there  is  no  rotation  of  the  lever,  it  is  obvious  that 
the  forces  tending  to  pro-        <       •$  n™ 

duce  a  clockwise  Q  rota-     ~T       T  jfflJT  T  -  T 

tion  are  just  balanced  by     /00ff    zog          |I         ioog    $og 
those  which  tend  to   ro- 
tate  the  bar  in  the  coun- 


ter-clockwise  Q  direc-  FlG>  54- 

tion.  On  computing  the  moments  of  the  forces,  we  find 
on  the  right-hand  side  50  x  40  and  100  x  25.  On  the  left 
we  find  20  x  25  and  100  x  40.  Now  if  the  moments  acting 
clockwise  be  added,  their  sum  will  be  found  equal  to  the 
sum  of  those  acting  counter-clockwise;  or, 

50  x  40  +  100  x  25  =  20  x  25  +  100  x  40. 
This  equation  illustrates  a  general  law  which  may  be 
stated  thus  : 

An  equilibrium  of  forces  results  when  the  sum  of  the  mo- 
ments which  tend  to  make  the  lever  rotate  in  one  direction 
equals  the  sum  of  the  moments  which  tend  to  make  it  rotate 
in  the  opposite  direction. 

The  mechanical  advantage  of  the  lever  can  also  be  found 

by  applying  the  prin- 
ciple of  work  set 
forth  in  §  89.  Let 
the  lever  shown  in 

FIG.  55.  -The  Work  Done  by  the  Effort  F       Fig-  55  turn   slightly 
Equals  That  Done  upon  the  Weight  W.         about    the    f  ulcrum    0 

until  it  has  the  position  cd.  Drawing  the  perpendiculars 
ac  and  bd,  we  have  from  similar  triangles 

ac  :  bd  :  :  Oc  :  Od. 
But,  since  Oc  equals  OA  and  Od  equals  OB, 

ac:bd  :  :  AO  :BO. 


jt 

O  ---•"""""" T 

I '        f 


92  A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

Now  the  work  done  by  the  effort  F  is  the  product  F  x  «<?, 
and  the  work  done  against  the  resistance  W  is  W  x  bd. 
Since  the  work  done  by  the  agent  is  equal  to  that  accom- 
plished by  the  lever, 

F  X  ac  =  W  X  bd,  or  W  :  F  :  :  ac  :  bd  :  :  AO  :  BO.          (5) 

Therefore,  the  mechanical  advantage  of  a  lever  is  the  ratio 
of  the  arm  AO  to  which  the  effort  is  applied  to  the  arm 
BO  on  which  the  resistance  acts. 


EXERCISES 

1.  Show  by  diagrams  the  relative  position  of  effort,  resistance,  and 
fulcrum  in  the  following  instruments  of  the  lever  type:  oars,  sugar 
tongs,  lemon  squeezer,  rudder  of  a  boat,  pitchfork,  spade,  can  opener, 
pump  handle. 

2.  Two  boys  weighing  respectively  60  and  45  Ib.  balance  on  oppo- 
site ends  of  a  board.     If  the  fulcrum  is  6  ft.  from  the  larger  boy,  how 
far  is  it  from  the  smaller  one  ? 

3.  The  arms  of  a  lever  of  the  first  class  are  3  ft.  and  7  ft.     What 
is  the  greatest  weight  that  a  force  of  60  Ib.  can  support? 

4.  A  lever  of  the  second  class  is  required  to  support  a  weight  of  500 
kg.;    the  effort  to  be  applied  is  only  50  kg.     If  the   bar  is  20   ft. 
long,  where  should  the  weight  be  attached  ? 

5.  Two   masses  weighing  respectively   10   and    15    kg.   balance 
when  placed  at  opposite  ends  of  a  bar  2   m.  long.     Where  is  the 
fulcrum  ? 

6.  The  short  arm  of  a  lever  is  30  cm.  long,  the  long  arm  270  cm. 

If  the  end  of  the  long 
arm  moves  1  cm.,  how  far 
will  the  end  of  the  short 
arm  move? 

7.  The  forearm  is 
raised  by  a  shortening 
of  the  biceps  muscle. 
Considering  the  forearm 
as  a  lever  whose  fulcrum 

is  at  the  elbow,  to  what 
FIG.  56. -Wagon  Scales.  clasg     doeg     it     belong? 


MACHINES 


93 


i 


What  is  gained,  force  or  speed?     When  the  arm  is  being  extended  in 
striking  a  blow,  to  what  lever  class  does  it  belong? 

8.  The  scales  .used  in  weigh- 
ing heavy  loads,  Fig.  56,  are  a 

series  of  levers  arranged  as  shown.       _______ 

Explain  how  a  small  weight  W 
can  balance  a  load  of  coal  weigh- 
ing a  ton  or  more. 

9.  Explain   how   a  parcel    is 
weighed  by  means  of  the  steel- 
yards shown  in  Fig.  57.  FlG-  57'  ~  Steelyards. 


4.    THE    PRINCIPLE    OF   THE   WHEEL   AND   AXLE 

96.    The  Wheel  and  Axle.  —  A  simple  form  of  the  wheel 
and  axle  is  shown  in  Fig.  58.    The  applied  force,  or  effort, 

acts  tangentially  to  the  rim  of  the 
wheel  B  through  a  cord,  thus  pro- 
ducing rotation.  As  the  wheel 
revolves,  the  cord  to  which  is  at- 
tached the  weight  W  is  wound  up 
around  the  axle  A,  and  the  weight 
is  thus  lifted. 

If  the  force  F  turns  the  wheel 
through  one  revolution,  the  dis- 
tance through  which  it  acts  is 
equal  to  the  circumference  of  the 
wheel,  or  2  TT^,  where  R  is  the  radius  of  the  wheel.  The 
work  done  by  the  agent  is  therefore  F  x  2  irR  (§  55).  One 
revolution  of  the  wheel  lifts  the  weight  W  a  distance 
equal  to  the  circumference  of  the  axle,  i.e.  to  2  Trr,  where  r 
is  the  radius  of  the  axle.  The  work  done  therefore  by 
the  machine  upon  the  weight  is  "FT  X  2  TTT.  Applying  the 
principle  of  work  as  stated  in  §  89, 


FIG.  58.  — The  Wheel 
and  Axle. 


whence  F  x  R  =  W  x  r,  or  W  :  F  :  :  R  :  r. 


(6) 


94 


A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


The  mechanical  advantage  of  a  wheel  and  axle  is  there- 
fore equal  to  the  radius  of  the  wheel  divided  by  the  radius 
of  the  axle.  Hence,  a  wheel  and  axle  may  be  used  to  over- 

R 

come  a  resistance   --  times  as  great  as  the  effort  applied  to 

the  circumference  of  the  wheel. 

It  should  be  observed  that  the  circumferences  of  the 
wheel  and  the  axle  bear  the  same  ratio  as  their  respective 
radii,  and  therefore  may  be  used  to  replace  R  and  r  in 
equation  (6). 

In  the  compound  windlass,  Fig.  59,  two  machines  of  the  wheel  and 

axle  type  are  combined.     The  effort  F  acting  along  the  circumference 

of  the  circle  described  by 
the  crank  produces  a  force 
that  is  transmitted  by 
means  of  a  small  cog 
wheel,  or  pinion,  to  the 
rim  of  the  second  wheel 
to  which  the  axle  is  at- 
tached. As  the  crank  is 
turned,  a  rope  is  wound 
around  the  axle,  thus  lift- 
ing a  weight  or  overcom- 
ing some  other  resistance. 
The  mechanical  advan- 
tage of  a  compound  wind- 
lass can  be  calculated  by 
resolving  the  machine  into 
simple  wheels  and  axles. 
The  mechanical  advan- 
FIG.  59.  —  A  Compound  Windlass,  or  Widen.  tage  of  the  compound 

machine  is  the  product  of 

the  mechanical  advantages  of  its  constituent  parts. 

In  many  instances  in  which  the  wheel  and  the  axle  are  compounded 

the  motion  is  transmitted  by  belts,  chains,  or  cables,  and  occasionally 

by  the  friction  of  the  circumferences. 

The  wheel  and  axle  and  the  pulley  may  be  looked  upon 
as  special  cases  of  the  lever.  Fig.  60  (1)  shows  that  the 


\ 


\ 


MACHINES 


95 


effort  F  acting  tangentially  to  the  rim  of  the  wheel  really 
acts  upon  a  lever  arm  equal  to  R,  the  radius  of  the  wheel. 
Similarly,  the  resist- 
ance W  acts  upon  a 
lever  arm  equal  to  the 
radius  of  the  axle. 
Applying  the  principle 
of  moments, 


\ 


F  x  R  =  W  x  r. 
The  case  of  a  single 


(3) 


FIG.  60. 


fixed  pulley  shown  in 
(2)  may  be  regarded  as  a  lever  of  the  first  class  whose  ful- 
crum is  at  0,  or  as  a  wheel  and  axle  of  which  the  radii  are 
equal.  Either  view  leads  to  the  relation  F=  W.  A  sin- 
gle movable  pulley  shown  in  (3)  is  similar  to  a  lever  of 
the  second  class,  the  fulcrum  being  at  0,  and  the  resistance 
at  the  center  as  shown.  Again,  applying  the  principle  of 
moments,  we  find  that  W  =  2  F  as  in  §  91. 

EXERCISES 

1.   Show  that  the    ordinary  kitchen    meat    chopper   and    coffee 
grinder  are  examples  of  the  wheel  and  axle.     Give  other  commonplace 

examples. 

2.  The  radius  of  a  wheel 
is  3  ft.,  and  that  of  its  axle 
12  in.    What  effort  would  be 
required   to   overcome  a  re- 
sistance of  600  Ib.  ? 

3.  How  much  work  would 
be  done  upon  the  resistance 
in  moving  it  a  distance  of  10 
ft.  ?    How  much  work  would 
have  to  be  done  upon   the 

FIG.  61.  — The  Capstan.  wheel? 

4.   The  arm  of  a  capstan,  Fig.  61,  measured  from  the  center,  is 
2  m. ;  the  radius  of  the  barrel  is  25  cm.     What  effort  would  be  re- 


96  A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

quired  to  produce  a  tension  of  500  kg.  in  the  rope  attached  to  the 
barrel? 

5.  What  is  the  mechanical  advantage  of  the  machines  in  Exercises 
2  and  4? 

6.  If  the  effort  were  applied  to  the  axle,  and  the  resistance  to  the 
wheel,  what  would  be  gained  by  using  a  wheel  and  axle?    Illustrate 
by  using  the  wheel  and  axle  described  in  Exer.  2. 

7.  The  pedal  of  a  bicycle  describes  a  circle  whose  radius  is  7 
in.     If  the  radius  of  the  attached  sprocket  wheel  is  3  in.,  find  the  pull 
on  the  chain  when  the  foot  pressure  is  45  Ib. 

8.  If  the  front  sprocket  of  a  bicycle  contains  21  teeth,  and  the  rear 
one  7,  how  far  will  one  turn  of  the  pedal  move  a  28-inch  wheel  along 
the  ground  ?    Find  the  number  of  turns  of  the  pedal  per  mile. 

9.  The  length  of  the  crank 
arm  shown  in  Fig.  62  is  10  in., 
and  the  radius  of  w  is  6  in.; 
wheel  w  is  belted  to  S,  whose 
radius  is  3  in.;  and  S  is  at- 
tached to  W,  whose  radius  is  20 

in.     One  turn  of  the  crank  pro- 
FIG.  62.  , 

duces  how  many  revolutions  of 

the  wheel  W'i     A  point  on  the  rim  of  W  moves  how  much  faster  than 
the  crank? 

10.  The  large  wheel  of  a  sewing  machine  is  12  in.  in  diameter, 
and  the  small  one  to  which  it  is  belted  is  3  in.  One  up-and-down 
movement  of  the  treadle  produces  how  many  stitches? 


5.    THE  INCLINED  PLANE,   SCREW,   AND  WEDGE 

97.  The  Inclined  Plane.  —  Let  AB,  Fig.  63,  represent 
an  inclined  plane  whose  surface  is  smooth  and  unbending. 
Let  the  height  BO  be  h,  and  the  length  AB  be  I.  If  the 
effort  F  acting  parallel  to  AB 
causes  the  ball  whose  weight  is 
W  to  move  the  distance  AB,  the 
work  done  by  the  agent  is  F  x  I. 
The  weight  W  is  lifted  a  dis-  ~ - 

tance  equal  to  BO,  and  the  WOrk       FIG>  63.  — The  Inclined  Plane. 


MACHINES 


97 


done  is  W  X  h.     According  to  the   general   principle   of 


Fxl=Wxh. 


(7) 


The  mechanical  advantage  —  is  therefore  equal  to  -,  i.e. 

F  h 

the  ratio  of  the  length  of  the  plane  to  its  height. 

98.  The  Screw.  —  The  screw  is  sometimes  considered 
to  be  a  modification  of  the  inclined  plane.  If  a  right  tri- 
angle, Fig.  64,  be  cut  from  paper  and 
wound  around  a  cylindrical  rod,  as 
shown,  the  hypotenuse  of  the  triangle 
forms  a  spiral  similar  to  the  threads' 
of  a  screw.  The  distance  between  two 
consecutive  turns  of  the  thread  measured 
parallel  to  the  axis  of  the  rod  is  the  FlG.  64.  _  A  Screw  is  a 

pitch  of  the  screw.  Spiral  Incline. 

The  equation  giving  the  mechanical  advantage  of  the 
screw  is  readily  derived  by  applying  the  principle  of  work 
(§  89)  as  follows  : 

Let  the  screw  shown  in  Fig.  65  be  turned  once  around 
by  applying  the  effort  F  to  the  end  of  arm  J.,  tangent  to 

the  circle  which  it  de- 
scribes. During  one 
revolution  the  effort 
acts  through  the  dis- 
tance 27JT,  where  r  is 
the  length  of  the  arm. 
The  work  done  is  F  x 
2  TIT.  While  the  screw 
is  making  one  revolu- 
FIG.  65. -The  Screw.  tion,  the  weight  W  is 

obviously  lifted  through  a  distance  equal  to  the  pitch  of 
the  screw,  which  may  be  represented  by  the  letter  s.     The 
work  done  by  the  screw  is  therefore  W  x  s.     Hence 
8 


98 


A  HIGH   SCHOOL  COURSE  IN   PHYSICS 


FIG.  66.— The 
Jackscrew. 


F  x  2  IT  x  r  =  W  x  s.  (8) 

The  mechanical  advantage  of  the  screw  is  therefore  the 

ratio  2  TTT/S. 

The  screw  as  a  mechanical  power  owes  its  importance 

to  the  fact  that  its  mechanical  advantage  can  be  enor- 
mously increased  simply  by  making  r  large 
and  8  small,  as  in  the  jackscrew,  Fig.  66, 
used  to  lift  buildings  from  their  founda- 
tions. It  is  extremely  useful  in  the  form 
of  bolts,  wood  screws,  vises,  clamps,  presses, 
water  taps,  and  in  many  other  cases  where 
a  great  multiplication  of  force  is  desired. 

99.    The  Wedge.  —  The  wedge  may  be  con- 
sidered as  two  inclined  planes  placed  base  to 

base.    It  is  used  for  splitting  logs  (Fig.  67),  raising  heavy 

weights  small  distances,  removing  the  covers  from  boxes, 

etc.      The    importance    of     the 

wedge  is  due  to  the  fact  that  the 

energy  imparted  to  it  is  delivered 

by  the   blows   of   a   hammer   or 

heavy    mallet.        Although    the 

amount  of  friction  to  be  overcome 

in  driving  a  wedge  is  necessarily 

very  large,  by  its  use  a  man  can 

overcome  enormous  resistances.     Since  friction  cannot  be 

disregarded,  no  definite  relation  between  effort  and  re- 
sistance can  be  given. 

EXERCISES 

1.  A  ball  weighing  10  Ib.  rests  upon  an  inclined  plane.     If  the 
height  of  the  plane  is  6  in.  and  the  length  is  30  in.,  what  effort  acting 
parallel  to  the  plane  will  be  required  to  hold  the  ball  in  equilibrium? 

2.  On  an  icy  slope  of  45°,  what  force  is  required  to  haul  a  sled  and 
load  weighing  a  ton,  neglecting  friction? 

3.  The  radius  of  the  wheel  of  a  letter  press  is  12  in.;  the  pitch  of 


FIG.  67.  —  One  Use  of  the 
Wedge. 


MACHINES  99 

its  screw,  \  in.     Neglecting  friction,  what  pressure  is  produced  by  a"n 
effort  of  50  lb.? 

4.  Neglecting  friction,  what  constant  force  must  a  team  of  horses 
exert  in  hauling  a  load  of  coal  weighing  3000  lb.  up  an  incline  of  30°? 

5.  What  is  gained  by  making  an  inclined  plane  of  a  given  height 
longer?    What  is  lost?    Illustrate  by  means  of  examples. 

6.  In  a  machine  the  effort  of  50  lb.  descends  20  ft.,  while  a  weight 
is  raised  10  in.     What  is  the  weight? 

7.  What  is  the  mechanical  advantage  of  a  screw  press  of  which 
the  pitch  of  the  screw  is  5  mm.   and  the  diameter  of   the  circle 
described  by  the  effort  50  cm.  ? 

8.  A  smooth  railroad  track  rises  50  ft.  to  the  mile.     A  car  weigh- 
ing 20  T.  would  require  how  much  force  to  keep  it  from  moving  down 
the  slope? 

6.  EFFICIENCY  OF  A  MACHINE 

100.  Friction  and  Efficiency.  —  On  account  of  the  fact 
that  there  is  always  a  resistance  due  to  friction  wherever 
one  part  of  a  machine  moves  over  another,  some  work 
must  be  done  in  moving  the  parts  of  the  machine  itself. 
The  useful  work  done  by  a  machine  is  therefore  less  than 
the  work  done  upon  it;   i.e.  W  X  d'  is  always  less  than 
F  X  d.      The  ratio  of  the  useful  work  done  by  a  machine  to 
the  work  done  upon  it  is  the  efficiency  of  the  machine. 

EXAMPLE.  —  In  the  working  of  a  pulley  system  an  effort  of  50  lb. 
acts  through  a  distance  of  20  ft.  and  lifts  a  weight  of  180  lb.  5  ft. 
What  is  the  efficiency  of  the  system  ? 

SOLUTION.  —  The  work  done  on  the  machine  is  50  x  20,  or  1000 
foot-pounds.  The  work  done  by  the  machine  is  180  x  5,  or  900  foot- 
pounds. The  efficiency  is  therefore  900  •«•  1000,  or  0.9. 

Efficiency  is  usually  expressed  as  a  percentage  of  the 
total  work  applied  to  a  machine ;  thus  in  the  example 
above  the  efficiency  is  90  %  • 

101.  Sliding    and    Rolling    Friction. — Since     friction 
always   tends   to   decrease    the    efficiency   of   machinery, 
advantage  is  taken  of  every  method  that  will  reduce  it 
to  the  smallest  possible  amount. 


100         A   HIGH   SCHOOL  COURSE  IN   PHYSICS 

Let  a  block  of  wood  or  metal  and  a  car  of  the  same  weight  be 
drawn  up  a  slight  incline  and  the  force  measured  by  a  dynamometer. 
It  will  be  found  that  the  car  is  more  easily  moved  than  the  block. 

Again,  place  a  piece  of  sheet  rubber  on  the  incline  under  the  car. 
A  greater  force  will  be  required  to  produce  motion  than  before. 

The  first  experiment  shows  clearly  a  great  difference 
between  the  sliding  friction  of  the  block  and  the  rolling 
friction  of  wheels  on  a  hard  surface.  The  second  experi- 
ment demonstrates  the  value  of  a  hard,  unyielding  road 
when  heavy  loads  are  to  be  drawn.  If,  however,  the 
wheels  are  provided  with  wide  tires,  they  sink  less  deeply 
into  the  roadbed  and  meet  with  less  resistance.  The  un- 
yielding surfaces  of  car  wheels  and  the  track  enable  a 
locomotive  to  pull  enormous  loads  on  account  of  the  small 
amount  of  rolling  friction. 

It  is  plain  that  in  the  axle  bearings  of  ordinary  vehicles 
the  moving  part  must  slide  over  the  stationary  part.     The 
friction  thus  brought  about  is  greatly  re- 
duced by  means  of  lubricating  oil,  but  is 
avoided   in   the   construction   of   bicycles, 
automobiles,  etc.,  by  substituting  ball  bear- 
ings, as  shown  in  Fig.  68.     In  this  way  the 
FIG.      68.  —  Ball  a  .  ,  .  J.. 

Bearings  of  a   moving  part  S  is  separated  from  the  station- 
Bicycle  Axle.      ary  part  A  by  balls  which  roll  as  the  wheel 
turns.     Thus  rolling  friction  takes  the  place  of   sliding 
friction. 

EXERCISES 

1.  In  what  way  are  we  dependent  upon  friction  in  the  process  of 
walking  ?     Why  do  we  encounter  difficulty  in  walking  on  smooth  ice  ? 

2.  Why  is  it  difficult  for  a  locomotive  to  start  a  train  on  wet  rails  ? 
How' is  the  difficulty  overcome? 

3.  Why  do  smooth  nails  hold  two  pieces  of  wood  together?    State 
other  ways  in  which  we  take  advantage  of  friction. 

4.  Calculate  the  efficiency  of  a  wheel  and  axle  when  an  effort  of 
20  Ib.  acting  through  30  ft.  lifts  a  weight  of  80  Ib.  7  ft. 


MACHINES  101 

5.  On  account  of  the  loss  of  energy  due  to  friction  in  a  pulley  sys- 
tem, an  effort  of  70  kg.  acting  through  30  m.  moves  a  resistance  of 
340  kg.  through  5.5  m.     What  is  the  efficiency  of  the  machine? 

6.  What  is  the  efficiency  of  a  screw  if  an  effort  of  5  kg.  applied  at 
the  end  of  an  arm  1  m.  long  produces  a  pressure  of  4000  kg.,  the  pitch 
of  the  screw  being  4  mm.  ? 

7.  The  efficiency  of  an  inclined  plane  is  50  %.     If  the  length  of  the 
plane  is  20  ft.  and  its  height  4  ft.,  what  effort  acting  parallel  to  the 
plane  will  be  required  to  move  a  body  weighing  500  Ib.  ? 

SUMMARY 

1.  The  simple  machines  are  the  pulley,  lever,  wheel 
and  axle,  incline  plane,  screw,  and  wedge.     Complicated 
machinery  is  made  up  of  simple  machines  (§  88). 

2.  The  work  done  by  an  agent  upon  a  machine  is  equal 
to  the  work  accomplished  by  it.     This  is  known  as  the 
Principle  of  Work.     Hence  a  machine  may  gain  force  at 
the  expense  of  distance  (or  speed),  or  it  may  gain  distance 
(or  speed)  at  the  expense  of  force  (§  89). 

3.  The  mechanical  advantage  of  a  machine  is  the  ratio 
of  the  resistance  overcome  by  it  to  the  effort  applied  to  it ; 
i.e.  TT:^(§90). 

4.  When  a  continuous  cord  is  used  in  a  system  of  pul- 
leys, the  equation  is  W  =  nF,  in  which  n  is  the  number  of 
parts  of  the  rope  supporting  the  movable  block  (§  91). 

5.  When  a  lever  is  used,  the  moment  of  the  effort  equals 
the  moment  of  the  resistance,  or  F  x  I  =  W  x  V  (§  93). 

6.  Levers   are  classified  according  to  the  relative  po- 
sition of  fulcrum,  effort,  and  resistance.     Lever  arms  are 
always  measured  from  the  fulcrum  on  lines  perpendicular 
to  the  acting  forces  (§  94). 

7.  When  several  forces  act  at  different   points   on   a 
lever,  the  sum  of  the  moments  tending  to  produce  a  clock- 
wise rotation  equals  the  sum  of  the  moments  tending  to 
produce  rotation  in  the  opposite  direction  (§  95). 


102          A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

8.  The  equation  of  the  wheel  and  axle   is  F  x  R  = 
W  x  r  (§  96). 

9.  The    equation   of   the   inclined    plane    is   F  x  I  = 
IT  x  A  (§97). 

10.  The    equation    of    the  screw    is    F   x   2  irr  =   W 
x  «  (§  98). 

11.  Friction  tends  to  reduce  the  work  accomplished  by 
a  machine.     The   efficiency  of  a  machine  is  the  ratio  of 
the  useful   work   done  by  a  machine  to  the  work  done 
upon  it  (§  100). 

12.  In  general  the  friction  of  sliding  parts  of  machinery 
is  greater  than  that  of  rolling  parts.     Friction  may  be  re- 
duced by  lubricants,  or  by  substituting  rolling  friction 
for   sliding   friction,   as   in   the   case   of   ball  and  roller 
bearings  (§  101). 


CHAPTER   VII 

MECHANICS   OF  LIQUIDS 

1.   FORCES  DUE   TO  THE   WEIGHT   OF  A   LIQUID 

102.  Pressure  of  Liquids  against  Surfaces.  —  When  a 
hollow  rubber  ball  or  a  piece  of  light  wood  is  forced  under 
water,  the  body  resists  the  action  of  the  force  submerging 
it  and  manifests  a  strong  tendency  to  return  to  the  surface. 
The  fact  that  some  bodies  float  on  water  and  other  liquids 
shows  also  that  there  exists  a  force  acting  against  the 
lower  surfaces  sufficient  to  counteract  their  weight. 

Let  a  lamp  chimney  against  the  end  of  which  a  card  has  been 
placed  be  forced  partly  under  water,  as  shown  in  Fig.  69.  The  card 
will  be  pressed  firmly  against  the  end  of  the 
chimney  by  a  force  acting  in  an  upward  di- 
rection, and  a  large  quantity  of  shot  or  sand 
may  be  poured  into  the  vessel  before  the  card 
is  set  free.  The  person  performing  the  ex- 
periment will  also  perceive  that  a  strong 
upward  force  acts  against  the  hand.  By 
lowering  the  chimney  to  a  greater  depth,  the 
upward  force  against  the  hand  will  be  in- 
creased and  a  larger  quantity  of  shot  or  sand 
will  be  required  to  free  the  card.  FIG.  69.  —  A  Liquid  Exerts 

an     Upward     Pressure 

103.  Relation  between   Force   and      against    the    Vertical 
Depth. — The  manner  in    which   the       Tube- 

force  exerted  by  a  liquid  against  a  surface  varies  with  the 
depth  may  be  experimentally  tested  by  means  of  a  gauge 
constructed  as  shown  in  Fig.  70. 

A  "thistle  tube"  having   a  stem    about   80   centimeters  long  is 
bent  at  an  angle  of  90  degrees  about  35  centimeters  from  one  end. 

103 


104          A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

A  piece  of  thin  sheet  rubber  is  tied  tightly  over  the  large  end  A .  If  a 
drop  of  ink  is  placed  in  the  tube  at  JB,  its  movements  along  the  tube 
will  indicate  changes  in  the  force  against  the 
rubber  surface  A.  Furthermore,  the  dis- 
tance the  drop  moves  when  A  is  submerged 
in  a  liquid  will  be  very  nearly  proportional 
to  the  applied  force. 

Let  the  gauge  be  clamped  in  the  position 
shown  in  the  figure,  and  let  a  graduated  lin- 
ear scale  be  attached  to  the  horizontal  tube 
containing  the  drop  of  ink.  Now  let  the  po- 
sition of  the  drop  be  read  upon  the  scale  and 
a  tall  vessel  of  water  brought  up  until  the  sur- 
face A  is  3  centimeters  beneath  the  free  sur- 
70.  —  The  Upward  ^ace  °^  tne  hquid.  Let  the  new  position  of 
Force  against  A  Varies  the  drop  be  read  and  its  displacement  com- 
with  the  Depth.  puted.  If,  now,  the  positions  of  the  drop  be 

read  after  submerging  the  surface  A  successively  to  the  depths  of  6,  9, 
and  12  centimeters,  the  movements  of  the  drop  will  be  found  to  be 
proportional  to  the  depths. 

Therefore,  as  shown  by  this  experiment,  the  upward 
force  exerted  by  a  liquid  of  uniform  density  against  a  given 
surface  is  directly  proportional  to  the  depth  of  that  surface. 

104.  Direction  of  Forces  at  a  Given  Depth.  —  Common 
experience  shows  that  liquids  press  against  surfaces  that 
are  vertical  or  oblique  as  well  as  horizontal.  In  every 
case  the  force  exerted  is  perpendicular  to  the  surface 
against  which  it  acts.  Thus  water  will  be  forced  through 
a  hole  in  the  side  of  a  pail  near  the 
bottom  as  well  as  through  one  in  the 
horizontal  bottom  itself.  A  compari- 
son of  the  forces  in  all  directions  at 
a  given  depth  may  be  made  by  modi- 
fying slightly  the  construction  of  the 
gauge  shown  in  Fig.  71  as  follows : 

_    Cut  the  glass  tube  about  1  centimeter 

from  the  bulb,  and  insert  a  piece  of  rubber        all  Directions. 


MECHANICS   OF  LIQUIDS 


105 


tubing  about  10  centimeters  long.  Now,  if  the  bulb  is  lowered  in  a 
large  vessel  of  water,  the  position  of  the  drop  will  not  change  when 
the  rubber  surface  is  turned  in  different  directions,  provided  the  depth 
of  its  center  is  kept  constant. 

This  experiment,  together  with  the  one  described  in 
§  103,  leads  to  the  following  conclusions : 

(1)  The  force  exerted  by  a  liquid  at  a  given  depth  is  the 
same  in  all  directions,  and 

(2)  The  force  exerted  by  a  liquid  in  any  direction  is 
directly  proportional  to  the  depth. 

105.  Pressures  Due  to  the  Weight  of  a  Liquid.  — From 
a  study  of  Fig.  72  it  may  readily  be  observed  that  the 
forces  exerted  by  liquids  against  surfaces 
are  due  to  the  weight  of  the  liquid,  i.e.  to 
gravity.  If  a  vessel  were  filled  with 
smooth  blocks  of  wood  of  .equal  size  and 
density,  block  2  would  suffer  a  downward 
force  equal  to  the  weight  of  block  1,  and 


9 

1 

5 

10 

2 

6 

11 

3 

7 

12 

4 

8 

would  in  turn  exert  an  equal  and  opposite  FIG.  72.  —  Forces 
reaction  upward  against  it.  Likewise,  3  Due  to  Gravity, 
must  support  the  weight  of  1  and  2,  4  must  support  -Z,  2, 
and  3,  etc.  In  each  instance  the  upward  reaction  is  equal 
to  the  downward  action.  Thus  it  is  clear  that  at  a  given 
surface  the  upward  and  downward  forces  are  equal,  and 
also  that  the  force  against  any  horizontal  surface  is  pro- 
portional to  the  depth  of  that  surface  below  the  upper 
surface  of  the  blocks  in  the  vessel* 

Now  liquids,  unlike  solids,  do  not  tend  to  keep  their 
form,  but  must  be  supported  on  the  sides.  Hence,  if  we 
imagine  the  vessel  in  Fig.  72  to  contain  a  liquid,  section 
4,  for  example,  will  press  sidewise  against  8  and  12,  whose 
reactions  back  against  4  preserve  equilibrium.  If  the 
liquid  were  without  weight,  section  4,  for  instance,  would 
suffer  no  crushing  force  due  to  the  weight  of  the  portions 


106 


A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


above  it,  and  consequently  would  exert  no  lateral  force 

against  the  portions  surrounding  it. 

106.    Total  Pressure  on  a  Given  Area.  —  If  the  experiment 

described  in  §  102  be  repeated,  and  water  substituted  for 

the  shot  or  sand 
(Fig.  73),  it  will 
be  found  that  the 
card  is  set  free 
from  the  end  of 
the  chimney  at  the 
instant  the  water 
on  the  inside 
reaches  the  height 
of  that  on  the  out- 
side. 

If  the  chimney  is 
cylindrical,  the 
downward  force 


FIG.  73.  —  Force  against  BC  is  Equal  to  the 
Weight  of  Column  ABCD. 


within  the  chimney  is  obviously  equal  to  the  weight  of  the 
column  of  water  ABCD.  If  the  area  of  the  end  of  the 
chimney  is  a  square  centimeters,  and  the  depth  h  centi- 
meters, the  volume  of  the  column  of  water  in  the  chimney 
is  ah  cubic  centimeters.  Since  the  density  of  water  is  1 
gram  per  cubic  centimeter  (§  9),  the  weight  of  the  column 
ABCD  is  ah  grains.  Hence  the  force  exerted  on  the  card 
is  ah  grams.  If  another  liquid  is  used  whose  density  is  d 
grams  per  cubic  centimeter,  the  weight  of  the  column,  and 
hence  the  force,  is  ahd  grams. 

The  following  rule  may  therefore  be  given: 

The  force  exerted  by  a  liquid  on  any  horizontal  surface  is 
equal  to  the  weight  of  a  column  of  the  liquid  whose  base  is 
the  area  pressed  upon,  and  whose  height  is  the  depth  of  this 
area  below  the  surface  of  the  liquid,  or  Force  =  ahd. 


MECHANICS   OF  LIQUIDS  107 

Since  the  force  exerted  by  a  liquid  against  a  surface  at 
a  given  depth  is  the  same  in  all  directions  (§  104),  the 
following  rule  for  computing  the  force  against  surfaces 
that  are  not  horizontal  is  often  employed : 

The  force  exerted  by  a  liquid  against  any  immersed  surface 
is  equal  to  the  weight  of  a  column  of  the  liquid  whose  base  is 
the  area  pressed  upon,  and  whose  height  is  the  distance  of  the 
center  of  mass  of  this  area  below  the  surface  of  the  liquid. 

In  the  English  system  the  area  should  be  expressed  in 
square  feet,  the  depth  in  feet,  and  the  density  of  the  liquid 
in  pounds  per  cubic  foot.  The  density  of  water  may  be 
taken  as  62.5  pounds  per  cubic  foot. 

EXAMPLE.  —  A  tank  4  ft.  deep  and  8  ft.  square  is  filled  with  water. 
Find  the  force  exerted  against  the  bottom  and  one  side. 

SOLUTION.  —  The  area  of  the  bottom  of  the  tank  is  8  x  8,  or  64  sq. 
ft.  Hence  the  volume  of  the  column  whose  weight  equals  the  force 
exerted  on  the  bottom  is  4  x  64,  or  256  cu.  ft.  The  force  against  the 
bottom  of  the  tank  is  therefore  256  x  62.5  lb.,  or  16,000  Ib. 

The  area  of  one  side  of  the  tank  is  4  x  8,  or  32  sq.  ft.  The  depth 
of  the  center  of  mass  of  the  side  is  2  ft.  Hence  the  volume  of  the 
column  of  water  whose  weight  equals  the  force  exerted  against  the 
side  is  2  x  32,  or  64  cu.  ft.  The  force  exerted  by  the  water  upon  this 
side  is  therefore  64  x  62.5  lb.,  or  4000  lb. 

The  term  pressure  should  be  confined  to  the  meaning  of 
force  per  unit  area*  The  pressure  at  a  given  point  is  ex- 
pressed in  terms  of  the  force  exerted  over  a  unit  area  at 
that  depth ;  for  example,  1000  grams  per  square  centi- 
meter, 15  pounds  per  square  inch,  etc. 

EXERCISES 

1.  Find  the  entire  force  exerted  against  the  bottom  of  a  rectangu- 
lar vessel  5x8  cm.  and  filled  with  water  to  a  depth  of  15  cm. 

2.  Find  the  pressure  per  square  foot  at  the  bottom  of  a  pond  10 
ft.  \\\  depth. 

3.  A  cylindrical  glass  jar  5  cm.  in  diameter  is  filled  to  a  depth  of 
15  cm.  with  mercury.     Find  the  force  against  the  bottom  and  the 


108          A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

pressure  per  unit  area.     The  density  of  mercury  is  13.6  g.  per  cubic 
centimeter. 

4.  A  tank  is  4  ft.  wide,  8  ft.  long,  and  3  ft.  deep.     Compute 
the  force  exerted  against  one  end  and  the  bottom  when  the  tank  is 
full  of  water. 

5.  At  a  depth  of  10  m.  of  sea  water,  what  is  the  pressure  in  grams 
per  square  centimeter?     (The  density  of  sea  water  is  1.026  g.  per  cubic 
centimeter.) 

6.  At  a  depth  of  25  ft.  of  sea  water,  what  is  the  pressure  per 
square  inch  ? 

SUGGESTION.  —  Find  first  the  force  exerted  on  a  surface  1  ft.  square 
at  the  given  depth. 

7.  A  cubic  inch  of  mercury  weighs  0.49  Ib.     Compute  the  force 
exerted  against  the  bottom  and  one  side  of  a  glass  tank  4  in.  wide,  6 
in.  long,  and  5  in.  deep  when  full  of  mercury. 

8.  Find  the  force  exerted  against  the  bottom  of  a  cubical  vessel 
whose  volume  is  1  liter  when  the  vessel  is  filled  with  mercury. 

9.  What  depth  of  water  will  produce  a  pressure  of  1  Ib.  per  square 
inch? 

10.  What  is  the  pressure  per  square  centimeter  at  the  bottom  of  a 
column  of  mercury  76  cm.  in  height?     (For  the  density  of  mercury 
see  Exer.  3.) 

11.  To  what  height  would  a  mercurial  column  be  supported  by  a 
pressure  of  1000  g.  per  square  centimeter? 

12.  A  gauge  connected  with  the  water  mains  of  a  city  showed  a 
pressure  of  65  Ib.  per  square  inch.     What  was  the  height  of  the  water 
in  the  standpipe  above  the  level  of  the  gauge  ? 

SUGGESTION.  —  Find  the  depth  of  water  required  to  produce  a  pres- 
sure of  65  Ib.  on  a  surface  of  1  sq.  in. 

13.  A  diver  is  working  at  a  depth  of  45  ft.     How  much  is  the 
pressure  per  square  inch  upon  the  surface  of  his  body? 

14.  A  rectangular  block  of  wood  is  placed  under  water  so  that  its 
upper  face,  which  is  8  x  10  cm.,  is  20  cm.  below  the  surface.     If  the 
thickness  of  the  block  is  4  cm.,  what  is  the  force  exerted  by  the  liquid 
against  each  of  its  faces? 

15.  How  much  is  the  force  against  a  dam  20  ft.  long  and  10  ft.  high 
when  the  water  rises  to  its  top  ? 

16.  A  hole  in  the  bottom  of  a  ship  which  draws  30  ft.  of  water  is 
temporarily  covered  with  a  piece  of  canvas.     How  much  is  the  pres- 
sure against  the  canvas  from  the  outside? 


MECHANICS   OF  LIQUIDS 


109 


17.  The  water  level  is  at  the  top  of  a  dam  30  ft.  high.     Compute 
the  pressure  per  square  foot  at  the  bottom  of  the  dam.     How  much  is 
the  pressure  halfway  down  ? 

18.  If  the  dam  in  Exer.  17  is  100  ft.  long,  how  much  is  the  total 
force  against  its  surface  ? 

107.  Pressure  in  Vessels  of  Different  Shapes.  —  It  may 

be  correctly  inferred  from  §  106  that  the  force  exerted  by 
a  liquid  against  a  given  area  does  not  depend  on  the  shape 
of  the  vessel  containing  the  liquid  used,  inasmuch  as  the 
computation  of  this  force  involves  only  the  area  and  depth 
of  the  surface  pressed  upon  and  the  density  of  the  liquid. 
This  fact  can  be  demonstrated  experimentally  as  follows  : 

Let  a  glass  funnel  be  selected  the  mouth  of  which  is  of  the  same  area 
as  the  end  of  the  lamp  chimney  used  in  §  106.  Place  a  card  across  the 
mouth  of  the  funnel  and  submerge  it,  as  shown 
in  Fig.  74.  The  card  will  be  pressed  against 
the  funnel  with  the  same  force  as  it  was  when 
the  chimney  was  used,  i.e.  with  a  force  equal 
to  the  weight  of  a  column  of  water  ABCD. 
If  water  is  now  poured  into  the  stem  of  the 
funnel  at  E,  the  card  will  become  free  precisely 
when  the  level  of  the  water  in  the  funnel  has 
reached  the  height  of  the  water  in  the  vessel 
outside.  In  this  condition  the  water  in  the 
funnel  exerts  the  same  force  downward  against 
the  card  as  the  water  on  the  outside  exerts 

in  an  upward  direction.  Hence  the  downward  force  is  equal  to  the 
weight  of  the  column  of  water  ABCD.  In  other  words,  the  force 
downward  against  the  card  is  exactly  the  same  as  it  was  when  the  cylindri- 
cal chimney  was  used. 

108.  The  Hydrostatic  Paradox.  —  An  apparent  contra- 

diction arises  when  we 
apply  the  laws  of  liquid 
pressure  to  vessels  of 
the  forms  shown  in  Fig. 
75.  Let  the  vessels  have 
FIG.  75.— The  Hydrostatic  Paradox.  bases  of  equal  size  and 


FIG.  74: — Equal  Down- 
ward Forces  in  Ves- 
sels of  Different 
Shapes. 


110 


A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


be  filled  to  the  same  depth  with  water.  In  each  case  the 
force  exerted  by  the  water  against  the  bottom  is  equal  to 
the  weight  of  the  liquid  column  ABOD.  Because  it  is 
apparently  an  impossibility  for  different  masses  of  a  liquid 
to  produce  the  same  pressure,  this  conclusion  is  often 
called  the  hydrostatic  paradox. 

An  application  of  the  laws  of  pressure  in  liquids  to  vessel  (3),  Fig. 
75,  will  show  how  the  total  pressure  on  the  bottom  BC  can  be  far 
greater  than  the  weight  of  liquid  contained  in  the  vessel.  Although 
the  total  pressure  on  the  surface  BC  is  equal  to  the  weight  of  a  column 
A  BCD  of  the  liquid  (§  107),  there  are  upward  forces  against  the 
surfaces  e/and  gh  equal  to  the  weight  of  the  liquid  that  would  be  re- 
quired to  fill  the  spaces  Aefa  and  bghD.  Hence  the  resultant  of  all 
the  forces  is  the  difference  between  the  downward  force  on  BC  and 
the  upward  forces  on  ef  and  gh.  This  is  obviously  the  weight  of  the 
liquid  in  the  vessel, 

109.  A  Liquid  in  Communicating  Vessels.  —  Let  tubes 
of  various  shapes  and  sizes  open  into  a  hollow  connecting 

arm,  as  shown  in  Fig.  76.  Any 
liquid  poured  into  one  of  the  tubes 
will  come  to  rest  at  the  same  level 
in  all.  Although  different  quan- 
tities of  the  liquid  are  present  in 
the  several  tubes,  yet  for  the  same 
depth  the  parts  are  in  equilibrium. 
The  explanation  is  as  follows  : 

Let  two  vessels  containing  water  be  in 
communication,  as  shown  in  Fig.  77.  Let  A  be  the  area  of  a  cross- 
section  of  the  connecting  tube.  The  force  tend- 
ing to  move  the  water  to  the  right  is  A hd, 
where  h  is  the  depth  of  the  center  of  the  area 
considered,  and  d  the  density  of  the  water. 
The  force  tending  to  move  the  liquid  to  the  left 
is  Ah'd.  These  two  forces  will  be  in  equilibrium 
only  when  h  and  Ji'  are  equal,  i.e.  when  the 
upper  surfaces  in  the  two  vessels  lie  in  the  same 
horizontal  plane. 


FIG.   76.  —  Liquid    Level   in 
Communicating  Vessels. 


FIG.  77.  —  A  Liquid  in 
Equilibrium. 


MECHANICS   OF  LIQUIDS 


111 


EXERCISES 

1.  A  cone-shaped  vase  has  a  base  of  100  cm.2  and  is  filled  with 
water  to  a  depth  of  45  cm.     Find  the  force  and  pressure  per  square 
centimeter  acting  on  the  bottom. 

2.  The  water  in  a  reservoir  supplying  a  city  is  150  ft.  above  an 
opening  made  in  a  pipe  being  laid  along  a  street.     Find  the  pressure 
in  pounds  per  square  inch  required  to  prevent  the  water  from  running 
out.  Ans.   65.1  Ib. 

3.  A  glass  tube  1  m.  long  is  rilled  with  mercury  (density  13.6  g. 
per  cubic  centimeter).     Find  the  pressure  against  the  closed  end  of 
the  tube  in  grains  per  square  centimeter  when  the  tube  is  (1)  vertical 
and  (2)  inclined  at  an  angle  of  45°. 

Ans.   1360  g.  per  square  centimeter. 
961.7  g.  per  square  centimeter. 

4.  A  column  of  water  is  lifted  25  ft.  in  a  pipe.     Calculate   the 
pressure  per  square  inch  that  it  exerts  against  the  bottom  of  the  pipe. 


2.  FORCE  TRANSMITTED  BY  A  LIQUID 


(1)  (2) 

Fia.  78.  — The  Multiplication  of  Force  by  a  Liquid. 

110.    Transmission  of  Pressure  —  Pascal's  Law.  —  Let  a 

vessel  of  the  form  shown  in  (1),  Fig.  78,  be  filled  with 
water  to  the  point  a.     A  pressure  will  be  exerted  on  every 


112 


A  HIGH   SCHOOL  COURSE  IN   PHYSICS 


square  centimeter  of  area  depending  on  the  depth  of  that 
area.  The  force  exerted  upward  against  the  shaded  area 
AB,  assumed  to  be  100  square  centimeters,  is  100  h  grams, 
if  h  is  the  depth  of  the  water  in  the  tube.  This  force  is 
entirely  independent  of  the  area  of  the  portion  of  the 
vessel  at  a.  Let  this  area  be  1  square  centimeter.  Now, 
if  1  cubic  centimeter  of  water  is  poured  into  the  vessel, 
the  depth  of  the  liquid  is  increased  1  centimeter,  and  the 
depth  of  the  surface  AB  becomes  h  +  1  centimeters.  The 
force  now  exerted  against  AB  is  100  (h  +  1)  grams, 
i.e.  each  square  centimeter  of  AB  receives  an  additional 
force  of  1  gram.  Hence,  the  force  exerted  on  a  unit  area  at 
a  is  transmitted  to  every  unit  area  within  the  vessel. 

This  fact  was  first  published  in  1663  by  Pascal,  a 
French  mathematician.  The  law  may  be  expressed  as 
follows : 

Force  applied  to  any  area  of  a  confined  liquid  is  trans- 
mitted undiminished  by  the  liquid  to  every  equal  area  of  the 
interior  of  the  containing  vessel  and  to  every  part  of  the 
liquid. 

111.  The  Hydraulic  Press. — In  the  discussion  of  Pas- 
cal's Law  given  in  the  preced- 
ing section,  it  makes  no  dif- 
ference whether  the  pressure 
added  to  the  small  area  a  is 
produced  by  a  gram  of  water 
poured  into  the  tube  or  ex- 
erted by  a  small  piston  fitting 
the  tube,  as  shown  in  (2), 
Fig.  78.  The  shaded  area  AB 
to  which  the  force  is  trans- 
mitted by  the  liquid  may  be 
FIG.  79. —AIL  Hydraulic  Press,  made  the  area  of  a  large  piston, 


MECHANICS  OF  LIQUIDS 


113 


as  shown.  In  this  case  a  force  of  1  gram  on  the  small 
piston  will  be  transmitted  to  each  square  centimeter 
of  the  large  one.  Hence  1  gram  on  the  small  piston  will 
balance  100  grams  placed  on  the  large  one.  Furthermore, 
any  effort  F  applied  to  the  small  piston  will  balance  a  re- 
sistance 100  times  as  great  as  itself.  Hence  a  mechanical 
advantage  (§  90)  of  100  is  secured.  The  hydraulic  press, 
a  machine  employed  in  factories  for  exerting  great  force, 
is  one  of  the  most  important  applications  of  Pascal's  Law. 
(See  Fig.  79.) 

The  hydraulic  press  is  analyzed  in  Fig.  80.  When  the  lever  L  is 
raised,  the  small  piston  P  is  lifted,  and  water  from  the  cistern  T 
enters  the  cylinder  B  through 
the  valve  v.  As  the  small  pis- 
ton is  forced  down,  v  closes, 
and  the  water  in  the  cylinder 
B  is  driven  past  the  valve  v' 
into  the  chamber  below  the 
large  piston  P'.  By  Pascal's 
Law,  the  force  exerted  again  st 
the  large  piston  P'  is  as  many 
times  that  applied  to  P  as 
the  area  of  P'  is  times  that 
of  P.  In  other  words,  the 
mechanical  advantage  is 
P'/P.  By  making  P  very 
small  and  P'  large,  any  de- 
sired mechanical  advantage 
may  be  secured.  Thus  hand 
presses  are  sometimes  used 


FIG.  80.  —  Sectional  Diagram  of  the 
Hydraulic  Press. 


that  are  capable  of  exerting  a  force  of  several  hundred  tons. 

112.  Principle  of  Work  and  the  Hydraulic  Press.  —  It 
may  be  observed  from  (2),  Fig.  78,  that  when  the  effort  F 
moves  the  small  piston  a  distance  of  1  centimeter,  the 
large  piston,  having  to  make  room  for  the  1  cubic  cen- 
timeter forced  below  it,  must  rise  -^o  °^  a  centimeter. 
9 


114          A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

Although  the  larger  force  is  100  times  as  great  as  the 
smaller,  it  acts  through  only  T^  as  great  a  distance. 
Hence  the  work  done  by  the  larger  piston  as  it  rises  is 
no  greater  than  the  work  done  on  the  smaller. 

113.  Artesian   or   Flowing    Wells.  —  The    tendency    of 
water  to  flow  from  a  point  of  higher  level  to  one  of  lower 
level  has  a  wide  application  in  artesian,  or  flowing,  wells. 
In   many   localities  there  are   formed   so-called   artesian 
basins  of  great  extent  in  which  a  stratum  of  porous  ma- 
terial, as  sand  or  other  substance  through  which  water 
can  pkss  with  comparative  ease,  lies  between   strata   of 
clay  or  rock  which  are  impervious  to  water.     At  distant 
points   these  layers   have  been  crowded   to   the    surface 
by  geologic  processes  where  the  porous   layer  has  been 
laid    bare,  thus  rendering    the  entrance    of    water    pos- 
sible.    Hence  when   borings   are    made    into    the   earth 
through  the  various  layers  and  into  the  porous  stratum, 
water     often    rises    to    the    surface   when    it    is    lower 
than    the    region    where    the    water  enters  the    porous 
layer. 

Deep  artesian  wells  exist  at  St.  Louis,  Mo. ;  Columbus, 
Ohio ;  Pittsburg,  Pa. ;  and  Galveston,  Texas.  Noted 
wells  are  found  at  Passy,  France  (1923  feet);  Berlin, 
Germany  (4194  feet);  Leipzig,  Germany  (5735  feet). 

114.  Supplying  Cities  with  Water.  —  Comparatively  few 
cities   are  so   favorably  situated   that   they  derive   their 
supply  of  water  from  mountain  springs.     The  output  of 
such  springs  is  collected  in  large  artificial  reservoirs  or 
lakes  from  which  it  is  piped  to  the  towns  where  it  is  dis- 
tributed in  the  usual  manner.     In  such  cases  the  elevation 
of  the  reservoir  is  such  as  to  produce  an  adequate  pressure 
for  all  ordinary  purposes.     The  pressure  may  be  estimated 
at  43.5   pounds   per   square    inch   for  every  100  feet  of 
elevation. 


MECHANICS   OF  LIQUIDS 


115 


In  towns,  however,  whose  location  is  less  favored  by 
nature,  the  water  from  springs  or  wells  must  be  elevated 
by  means  of  pumps  (§  156)  to  suitable  reservoirs  or  tanks 
constructed  upon  high  ground  from  which  it  is  distrib- 
uted through  pipes  to  the  consumers.  In  some  large 
cities  no  reservoir  is  used,  the  pumping  engines  being  so 
adjusted  as  to  supply  the  water  at  a  given  pressure  as 
fast  as  it  is  consumed. 

115.  Water  Motors.  —  In  cities  where  water  is  delivered 
through  pipes  under  sufficient  pressure,  its  power  may 
be  utilized  in  running  sewing  ma- 
chines, polishers,  lathes,  etc.,  by 
employing  a  rotary,  water  motor. 
A  common  type  is  shown  diagram- 
matically  in  Fig.  81.  Water  is- 
sues with  great  velocity  from  the 
jet  J  against  cup-shaped  fans  at- 
tached to  the  axle  of  the  motor. 
These  are  inclosed  in  a  metal  case  FIG.  81.— Sectional  View  of  a 
O  from  which  the  water  flows  into  Water  Motor- 

the  sewer  or  other  drain.     The  impact  of  the  water  against 

the  fans  suffices  to  turn 
the  shaft  to  which  are 
connected  either  di- 
rectly or  by  belts  the 
machines  that  it  is  de- 
sired to  operate. 

116.  Water  Wheels. 
—  An  elevated  body  of 
water,  like  a  lifted 
weight,  is  a  source  of 

FIG.  82.  —  An  Overshot  Water  Wheel.  ,.   ,  T,    . 

potential  energy.     It  is 

the  falling  of  this  water  to  a  lower  level  that  supplies  the 
power  for  operating  innumerable  mills,  factories,  electric 


116 


A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


power  plants,  etc.,  found  throughout  this  and  other  coun- 
tries. In  order  to  enable  water  to  do  the  required  work, 
the  so-called  overshot  water  wheel  has  long  been  employed. 
A  wheel  of  this  kind  is  shown  in  Fig.  82.  It  is  plain  that 
the  impact  and  weight  of  the  moving  water  from  above 
the  wheel  conspire  to  turn  the  wheel,  to  which  is  geared 
or  belted  the  machinery  to  be  operated.  Another  old  but 
common  form  of  water  wheel  is  the  undershot  wheel.  In 
this  case  the  wheel  is  turned  by  the  impact  of  the  current 
of  water  against  fans  at  the  bottom. 

The  most  efficient  form  of  water  wheel,  however,  is  the 
turbine,  which  is  utilized  in  all  modern  plants  employing 

water    power.       Water 

is  conducted   from   the 

+*~~  i  '""*•— ^  „'" 

£    c  r     ^tt  "^     -x'"'^'          reservoir  above  a  dam 

through  a  closed  cylin- 
drical tube,  or  flume, 
F,  Fig.  83,  to  a  penstock 
(shown  by  the  dotted 
lines)  which  surrounds 
the  stationary  iron  case 
T  containing  the  rotat- 
ing wheel,  or  turbine. 
This  case  rests  upon  the 
floor  of  the  penstock 
and  is  submerged  in  water  to  a  depth  equal  to  the  "  head," 
or  height,  of  the  water  supply.  The  turbine  is  attached 
to  the  shaft  8  and  is  set  in  rotation  when  the  water  is  ad- 
mitted. The  small  shaft  P  serves  to  control  the  size  of  the 
openings  in  the  case  through  which  the  water  gains  entrance 
to  the  turbine.  Figure  84  is  a  sectional  view  through  the 
turbine  R  and  the  case  8.  Water  enters  as  shown  by  the 
arrows  and  strikes  the  blades  of  the  turbine  at  the  most 
effective  angle  for  producing  rotation.  When  the  water 


FIG.  83.  —  A  Water  Turbine. 


MECHANICS   OF  LIQUIDS 


117 


FIG.  84.  —  Section  of  a  Water 
Turbine. 


has  expended  its  energy,  it  falls  from  the  bottom  of  the 

case  into  the  tailrace  below  the  penstock.     The  efficiency 

(§  100)  of  turbines  is  frequently 

as  high  as  85  or  90  per  cent. 
117.    Hydraulic  Elevators.  — 

Another  use  made  of  the  energy 

of  water  under  pressure  is  in  the 

operation  of  elevators,  or  lifts. 

One  form  is  shown  in  Fig.  85. 

When  water  is  admitted  to  the 

cylinder  O  through  the  control- 
ling valve  F",  the  piston  is  forced 

to  the  right,  thus  producing  a 

pull  upon  the  cable  (shown  by  the  dotted  lines)  and  caus- 
ing the  car  A  to  ascend.  Check- 
ing the  flow  of  water  by  means  of 
the  rope  r  stops  the  car,  while 
turning  valve  V  to  the  position 
shown  in  the  figure  allows  the 
water  to  flow  from  the  cylinder 
and  causes  the  car  to  descend  by 
its  own  weight.  A  study  of  the 
figure  will  show  that  the  car  moves 
four  times  as  fast  and  four  times 
as  far  as  the  piston. 

In  another  form  the  piston  is  a 
long  vertical   cylinder   extending 
from  the  bot- 
of     the 


1  '..•>  s     .  .  >  •>  '. 


ii     ii    ii 


Mf 


deep    hollow 

FIG.  85.  —  An  Hydraulic  Elevator.  C vlinder      set 

in  the  ground.     The  admission  of  water  under  pressure 
acting  against  the  lower  end  of  the  long  piston  lifts  the 


118 


A  HIGH   SCHOOL  COURSE  IN   PHYSICS 


car.     The  method  of  controlling  such  elevators  is  the  same 

as  that  shown  in  Fig.  85. 

118.    The  Hydraulic  Ram.  —  The   hydraulic   ram   is  a 

useful  device  for  automatically  elevating  water  in  small 

quantities  when  an 
abundant  supply  of 
spring  water  is  available 
which  has  a  fall  of  only 
a  few  feet.  Water  flows 
from  the  source  through 
FIG.  86.— The  Hydraulic  Ram.  the  pipe  P,  Fig.  86,  and 

out  through  the  opening  at  v.  As  the  water  increases  in 
speed,  the  valve  at  v  is  lifted,  which  causes  a  sudden  inter- 
ruption of  the  flow  in  P.  Since  the  momentum  of  the 
water  cannot  be  destroyed  instantaneously,  a  portion  of  the 
water  is  driven  forcibly  past  valve  vf  into  the  air  chamber 
0.  A  slight  rebound  of  the  water  in  the  large  pipe  re- 
lieves for  a  moment  the  pressure  against  valve  v,  which 
falls  by  its  own  weight,  thus  opening  again  the  orifice  at 
that  point.  A  repetition  of  the  process  causes  more  water 
to  enter  O  until  finally  the  pressure  is  sufficient  to  lift  a 
portion  of  the  water  to  a  height  of  many  feet.  The  com- 
pressed air  in  Q  acts  as  an  elastic  cushion  and  serves  also 
to  keep  a  steady  flow  of  water  in  the  small  pipe.  The 
quantity  of  water  delivered  by  an  hydraulic  ram  is  de- 
pendent on  the  height  of  the  source,  the  elevation  to 
which  it  is  to  be  lifted,  the  friction  of  the  pipes,  the 
length  of  pipe  P,  and  the  size  of  the  ram  itself. 

EXERCISES 

1.  The  area  of  the  small  piston  of  an  hydraulic  press  is  2  cm.8 
and  that  of  the  large  one  80  cm.2      How  much  force   will  50   Kg. 

applied  to  the  former  produce  upon  the  latter  ? 

2.  The  small  piston  of  an  hydraulic  press  is  operated  by  a  lever  of 
the  second  class  4  ft.  in  length,  and  the  piston  rod  is  attached  12  in. 


MECHANICS  OF  LIQUIDS  119 

from  the  fulcrum.     If  the  diameters  of  the  pistons  are  1  in.  and  8  in. 
respectively,  how  great  an  effort  will  produce  a  force  of  2  T.  ? 

3.  If  the  effort  applied  to  the  small  piston  in  Exer.  2  moves  through 
1  ft.,  how  much  will  the  large  piston  be  raised  ? 

4.  A  piston  moves  in  a  cylinder  that  is  in  communication  with  a 
water  system  whose  pressure  is  65  Ib.  per  square  inch.     If  a  force  of 
1  T.  is  to  be  developed  by  the  piston,  what  is  the  least  diameter  that 
it  can  have?  Ans.  6.26  in. 

5.  Pressure  against  a  piston  20  cm.  in  diameter  is  produced  by  a 
column  of  water  30  m.  high.     Calculate  the  force  against  the  piston 
and  the  work  performed  when  the  piston  moves  4  m. 

6.  Give  suitable  dimensions  to  the  pistons  and  lever  of  an  hydraulic 
press  in  order  that  an  effort  of  1  Ib.  may  produce  a  force  of  3000  Ib. 

.  ARCHIMEDES*   PRINCIPLE 

119.  Buoyancy  of  Liquids.  —  It  is  a  matter  of  common 
observation  that  bodies  apparently  become  lighter  when 
placed  under  water.     If  the  hand,  for  example,  be  sub- 
merged in  a  vessel  of  water,  it  becomes  evident  at  once 
that  it  is  supported  almost  without  muscular  effort.     Let 
a  one-  or  two-pound  stone  be  weighed  in  air  and  then 
weighed  again  while  immersed  in  water.     A  decrease  of 
several  ounces  will  be  observed.     When  blocks  of  wood  or 
many  other  bodies  are  placed  in  water,  the  buoyancy  of 
the  water  is  sufficient  to  cause  them  to  float. 

120.  The  Principle  of  Archimedes.1 —  On  account  of  the 
importance  of  the  laws  relating  to  the  apparent  decrease 

1  ARCHIMEDES  (287-212  B.C.).  The  name  of  Archimedes  will  be  re- 
membered by  students  of  Physics  in  connection  with  the  buoyant  action 
of  liquids  on  immersed  solids.  The  story  is  related  that  Hiero,  King  of 
Syracuse,  had  ordered  a  crown  of  pure  gold  which,  when  delivered,  al- 
though it  was  of  the  proper  weight,  was  suspected  to  contain  a  quantity  of 
silver.  Archimedes  was  asked  to  investigate.  A  method  of  procedure 
occurred  to  him  while  in  the  public  bath  as  he  noticed  that  his  body  ex- 
perienced a  greater  buoyancy  the  more  completely  it  was  submerged. 
Recognizing  in  this  effect  the  key  to  the  solution  of  the  problem,  he 
leaped  from  the  bath  and  hurried  homeward,  exclaiming  "Eureka! 


120          A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

in  weight  that  bodies  undergo  when  submerged  in  a  liquid 
three  experiments  will  be  described: 

1.  Measure  the  dimensions  of  a  metal  cylinder  or  rectangular 
metal  block,  and  compute  its  volume  in  cubic  centimeters.  From  the 

volume  compute  the  weight  of  an 
equal  volume  of  water.  Suspend 
the  metal  body  from  one  arm  of  a 
balance,  and  ascertain  its  weight  in 
air.  Weigh  the  body  also  when  sub- 
merged in  water,  and  compute  the 
loss  of  weight.  Compare  the  loss 
of  weight  with  the  weight  of  an 
equal  volume  of  water  found  at  first. 
2.  From  one  arm  of  a  balance, 
Fig.  87,  suspend  a  metal  cylinder  A 
and  the  bucket  B  whose  capacity  is 
precisely  equal  to  the  volume  of 
FIG.  87. -Verifying  Archimedes'  the  cylinder.  Counterbalance  these 
Principle.  ,  ,  »'•-•'*, 

by  means  of  sand  or  weights  placed 

in  the  opposite  scale  pan.  Submerge  A  in  water,  and  equilibrium  will 
be  destroyed.  Fill  the  bucket  with  water,  and  equilibrium  will  be 
restored  to  the  system. 

3.  Place  a  small  glass  beaker  (or  tumbler)  on  one  pan  of  a  balance, 
and  suspend  a  stone  from  beneath  the  same  pan.  Counterpoise  by 

Eureka!"  which  means  "I  have  found  it!"  Experiments  showed 
that  equal  masses  of  gold  and  silver  weigh  unequal  amounts  when  sub- 
merged in  water.  Therefore,  when  the  crown  was  weighed  against  an 
equal  mass  of  pure  gold,  both  being  submerged,  the  fraud  was  at  once 
detected. 

Archimedes  is  regarded  as  the  founder  of  the  science  of  Mechanics. 
From  his  time  nearly  2000  years  elapsed  before  any  great  advance  was 
made.  His  investigation  of  levers  is  noteworthy.  He  is  said  to  have  made 
this  remark,  "Give  me  a  fulcrum  on  which  to  rest  my  lever  and  I  will 
move  the  earth."  Among  his  countrymen,  he  was  probably  best  known 
on  account  of  the  numerous  instruments  of  war  which  he  invented. 

As  a  mathematician,  Archimedes  was  first  to  determine  the  value  of  IT, 
and  the  nrst  to  compute  the  area  of  a  circle.  He  is  supposed  to  have  been 
killed  by  a  Roman  soldier  while  engaged  in  the  investigation  of  a  problem 
in  geometry. 


MECHANICS   OF  LIQUIDS  121 

using  sand,  shot,  or  weights.  Next  fill  a  vessel  provided  with  a  spout 
with  water,  and  allow  all  the  excess  to  flow  out  at  the  spout.  Place  an 
empty  tumbler  under  the  spout,  and  lower  the  counterpoised  stone 
into  the  water.  Completely  submerge  the  stone,  and  catch  all  the 
displaced  water,  the  volume  of  which  will  be  equal  to  that  of  the 
stone.  Pour  the  displaced  water  into  the  beaker  on  the  scale  pan,  and 
equilibrium  will  be  restored. 

From  the  results  obtained  by  making  any  one  of  the 
experiments  just  described  Archimedes'  Principle  may 
be  deduced: 

A  body  immersed  in  a  liquid  is  buoyed  up  by  a  force  equal 
to  the  weight  of  the  liquid  that  it  displaces. 

121.  Explanation  of  Archimedes'  Principle.  —  The  rea- 
sons for  Archimedes'  law  become  clear  when  the  laws  of 
liquid  pressure  stated  in  §  104  are  applied 
to  a  submerged  body.  Let  a  rectangular 
block  abed  be  immersed  in  a  liquid,  as 
shown  in  Fig.  88.  The  force  against  the 
upper  surface  of  the  block  is  the  weight 
of  the  column  of  the  liquid  efad  and  acts 
downward.  The  force  exerted  by  the 
liquid  against  the  lower  surface  of  the 


block  cb  is  the  weight  of  a  column  of  the  FlG-   88.  — The   Up- 

T        •  i        /.T  i        mi  ward     Force    Ex- 

liquid  efbc  and  acts  upward.  The  re-  Ceeds  the  Down- 
sultant  of  these  two  forces  is  obviously  ward  Force- 
in  an  upward  direction  and  equal  to  the  weight  of  a  volume 
abed  of  the  liquid.  The  lateral  forces  exerted  by  the  liquid 
against  any  two  opposite  faces  of  the  block  are  equal  and 
opposite,  and  therefore  produce  no  tendency  to  move  the 
block.  Hence  the  immersed  body  is  buoyed  up  by  a  force 
equal  to  the  weight  of  a  volume  of  the  liquid  equal  to  its 
own  volume,  i.e.  to  its  displacement.  It  should  tfe  ob- 
served that  the  depth  to  which  the  body  is  submerged  does 
not  affect  the  final  result. 


122          A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

If  the  weight  of  the  body  submerged  in  the  liquid  just 
equals  the  weight  of  the  displaced  liquid,  the  body  will  be 
in  equilibrium  and  remain  where  it  is  placed  provided  the 
density  of  the  medium  be  uniform.  If  the  body  weighs 
more  than  the  amount  of  liquid  displaced,  it  will  sink;  if 
less,  it  will  rise  to  the  surface  and  float. 

122.  Floating  Bodies.  —  1.  Obtain  a  square  stick  of  pine  about 
30  centimeters  long  and  1  centimeter  square.  Measure  its  dimensions 

carefully,  and  beginning  at  one  end, 
lay  off  cubic  centimeters  after  taking 
due  account  of  the  cross-sectional 
area.  Drill  a  deep  hole  in  the  end  of 
the  bar  and  embed  a  nail  heavy  enough 
to  cause  the  bar  to  float  in  an  upright 
position,  as  shown  in  (1)  Fig.  89. 
Melt  paraffin  into  the  pores  of  the 
wood  over  a  flame  in  order  to  make  it 
water-proof.  Float  the  bar  in  water, 

and  read  off  the  number  of  cubic  cen- 
FIG.   89.  —  Flotation  Illustrated. 

timeters  of  the   portion  beneath  the 

liquid  surface,  i.e.  the  displacement.     Compare  the  weight  of  the  water 
displaced  with  the  weight  of  the  bar  itself. 

2.  Counterpoise  a  glass  beaker  (or  tumbler)  on  a  balance.  Prepare 
a  vessel,  as  shown  in  (2)  Fig.  89,  to  catch  displaced  water.  Carefully 
float  a  piece  of  well-paraffined  wood  weighing  about  100  grams,  and 
catch  the  displaced  water.  Wipe  the  block  of  wood  dry,  and  place  it 
on  one  scale  pan  of  the  balance,  and  pour  the  displaced  water  into  the 
beaker  placed  on  the  other.  The  two  should  balance. 

If  a  block  of  wood  is  entirely  immersed  in  water,  the  dis- 
placed water  weighs  more  than  the  wood,  and  therefore  the 
buoyant  force  is  greater  than  the  weight  of  the  block. 
Hence  the  block  will  be  forced  toward  the  surface.  As  a 
portion  of  the  block  rises  above  the  surface  of  the  water, 
the  displacement  decreases  until  the  weight  of  the  block 
and  that  of  the  displaced  water  are  equal.  This  fact  will 
be  found  to  be  verified  by  either  of  the  experiments  just 
described.  The  law  may  be  expressed  as  follows: 


MECHANICS   OF  LIQUIDS 


123 


A  floating  body  sinks  to  such  a  depth  in  the  liquid  that  the 
weight  of  the  liquid  displaced  equals  the  weight  of  the  body. 

123.    Measuring  Volumes  by  Archimedes'   Principle. — 

Archimedes'    principle    affords    an    easy    and    accurate 

method  for  ascertaining  the  volumes 

of  solid  objects  of  irregular  shapes. 

A  body  immersed  in  water  evidently 

displaces  a  volume  of  water  equal  to 

its   own   volume,  and,  according   to 

Archimedes'  principle,  loses  in  weight 

an  amount  equal  to  the  weight  of  the 

displaced  volume  of  water.    (See  Fig. 

90. )   Since  1  cubic  centimeter  of  water 

weighs  1  gram,  the  body  immersed 

contains  as  many  cubic  centimeters  as 

the   number   expressing   its    loss    of 

weight   in   grams.     For   example,  a 

piece   of    metal    that   weighs    45.75 

grams  in  air  and  40.23  grams  when 

immersed  in  water  displaces  the  difference,  or  5.52  grams 

of  water.     The  volume  of  water  displaced  is  therefore  5.52 

cubic  centimeters.     Hence  the  volume  of  the   immersed 

body  is  5.52  cubic  centimeters.     When  the  loss  of  weight 

is  measured  in  pounds,  the  volume  expressed  in  cubic  feet 

is  found  by  dividing 
that  loss  by  62.5. 
Why? 

124.  The  Floating  Dry 

Dock.  —  Among  the  useful 
modern  inventions  depend- 
ing upon  the  law  of  flotation 


FIG.  90.  —  Weighing  a 
Body  in  Water  to  Find 
its  Volume. 


FIG.  91.  —  The  Floating  Dry  Dock. 


(J  122)  is  the  floating  dry  dock,  which  is  shown  diagrammatically 
in  Fig.  91.  When  the  several  water-tight  compartments  P,  P,  P  are 
allowed  to  fill  with  water,  the  dock  sinks  until  the  water  level  is  at 


124          A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

AB.  A  vessel  to  be  repaired  is  then  floated  into  the  dock,  and  the 
water  pumped  out  of  the  various  compartments.  As  the  chambers 
are  emptied  the  dock  rises  sufficiently  to  bring  the  water  level  to 
the  line  CD.  The  boat  is  thus  lifted  clear  of  the  water. 

EXERCISES 

1.  A  stone  weighing  400  g.  under  water  weighs  480  g.  in  air.  What 
mass  and  volume  of  water  does  it  displace?    What  is  the  volume  of 
the  stone? 

2.  What  is  the  volume  of  a  metal  cylinder  that  weighs  30  g.  in  air 
and  19  g.  when  immersed  in  water? 

3.  A  solid  weighs  20  Ib.  in  air  and  12  Ib.  when  suspended  under 
water.     What  is  the  weight  of  an  equal  volume  of  water?    What  is 
the  volume  of  the  body  in  cubic  inches? 

4.  A  body  weighing  50  g.  in  air  weighs  35  g.  when  immersed  in 
water  and  38  g.  when  immersed  in  oil.     Find  the  mass  and  volume 
of  the  oil  displaced. 

5.  A  block  of  iron  weighing  12  g.  and  a  piece  of  wood  weighing 
4  g.  are  fastened  together  and  weighed  in  water ;  their  weight  when 
immersed  is  7.5  g.     If  the  iron  alone  weighs  10.2  g.  when  immersed 
in  water,  what  is  the  volume  of  the  wood? 

SUGGESTION.  —  From  the  combined  volumes  subtract  that  of  the 
iron. 

6.  A  block  of  wood  is  floated  in  a  vessel  full  of  oil.     If  200  g.  is 
the  weight  of  the  oil  displaced,  what  is  the  weight  of  the  wood? 

7.  A  boat  that  weighs  450  Ib.  displaces  how  many  cubic  feet  of 
water? 

8.  A  ferry-boat  weighing  700  tons  takes  on  board  a  train  weigh- 
ing 550  tons.     Express  the  total  displacement  in  cubic  feet. 

9.  Why  does  throwing  the  hands  out  of  water  cause  the  head  of  a 
swimmer  to  be  submerged  ? 

10.  An  egg  will  sink  in  water  and  float  in  brine.     A  solution  of 
salt  may  be  made  of  such  a  strength  that  an  egg  will  remain  at  any 
depth.     Explain. 

11.  What  is  the  volume  of  a  man  weighing  150  Ib.  if  he  floats  with 
2^  of  his  body  above  water  ? 

4.   DENSITY  OF  SOLIDS  AND  LIQUIDS 

125.    Density  of  a  Solid.  —  (#)    When  the  solid  is  more 
dense  than  water.     The  density  of  a  body  is  defined  as  its 


MECHANICS   OF  LIQUIDS 


125 


mass  per  unit  volume  and  is  found  by  dividing  the  number 
of  units  of  mass  by  the  number  of  units  of  volume  (§  12). 
In  the  C.  G.  S.  system  density  is  expressed  in  grams  per 
cubic  centimeter.  For  example,  the  density  of  mercury  is 
13.59  grams  per  cubic  centimeter. 

In  ascertaining  the  density  of  a  solid  that  sinks  in  water 
and  does  not  dissolve,  the  mass  is  first  found  by  weighing 
the  body  in  air,  and  the  volume  is  then  measured  by  the 
method  described  in  the  preceding  section.  By  definition, 

mass  (in  grams)  ,  >. 

volume  (in  cm.'6) 

But,  if  the  number  of  units  of  volume  is  found,  as  in 
practice,  by  ascertaining  the  apparent  loss  of  weight  sus- 
tained when  the  solid  is  submerged  in  water,  we  have  for 
the  numerical  value  of  the  density  in  the  metric  system, 

mass  (in  grams) . 


Density  (in  grams  per  cm.s) 


Density  = 


(2) 


weight  lost  in  water  (in  grams) 

(b)  When  the  solid  is  less  dense  than  water.  When  a 
solid  is  not  dense  enough  to  sink  in  water,  it  may  be 
attached  to  a  sinker  that  ._ 

is  sufficiently  heavy  to 
submerge  it.  Let  a 
sinker  S  and  a  light 
solid  A  whose  mass  is 
M  grams  be  suspended 
from  the  arm  of  a  bal- 
ance, as  shown  in  Fig. 
92,  so  that  the  sinker 
alone  is  submerged. 

Let    the    weight    of    the    FIG.  92.  —  Ascertaining  the  Volume  of  a 
i     j-      41  -,  Solid  Less  Dense  than  Water. 

two  bodies  thus  arranged 

be  W1  grams.     Now  let  both  solids  be  immersed,  and  the 

combined  weight  be  W2  grams.     The  difference  Wl  —  TF2 


126          A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

is  evidently  caused  by  the  buoyant  force  of  the  water  on 
the  light  solid  A,  and,  according  to  Archimedes'  principle, 
is  equal  to  the  weight  of  the  water  displaced  by  this 
body.  This  difference  is  numerically  equal  to  the  volume 
of  the  body  expressed  in  cubic  centimeters.  Hence  an 
equation  expressing  the  numerical  value  of  the  density  of 
the  solid  may  be  written  as  follows  : 

Density  =       *  (™  gram*) (3) 

wi  ~  W2  (m  grams) 

126.  Specific  Gravity. —  Occasional  use  is  made  of  the 
term  specific  gravity  (abbreviated  sp.  gr.)  to  express  the 
heaviness  or  lightness  of  a  body  as  compared  with  the 
weight  of  some  standard  substance.  The  specific  gravity 
of  any  solid  or  liquid  is  the  ratio  of  the  weight  of  the  body  to 
the  weight  of  an  equal  volume  of  pure  water  at  4°  0.  In  the 
case  of  gases  the  standard  with  which  they  are  compared 
is  either  air  or  hydrogen. 

Since  one  cubic  centimeter  of  pure  water  at  4°  C.  weighs 
one  gram,  it  follows  that  the  density  of  a  body  in  grams 
per  cubic  centimeter  is  numerically  equal  to  its  specific 
gravity.  For  example,  the  density  of  lead  is  11.36  grams 
per  cubic  centimeter;  hence  lead  is  11.36  times  as  heavy 
as  an  equal  volume  of  water.  Therefore,  the  specific 
gravity  of  lead  is  11.36. 

Again,  since  the  specific  gravity  of  lead  is  11.36,  one 
cubic  foot  of  lead  will  weigh  62.5  pounds  x  11.36,  or  710 
pounds.  Hence  in  the  English  system  of  measurement 
the  density  of  lead  is  710  pounds  per  cubic  foot. 

It  is  to  be  carefully  observed  that  specific  gravity  and 
density  are  not  the  same  thing ;  they  are  numerically  equal 
only  when  density  is  given  in  C.  G.  S.  units.  In  the 
English  system,  density  is  entirely  different  from  specific 
gravity,  as  the  example  plainly  shows. 


MECHANICS  OF  LIQUIDS  127 

The  equation  of  specific  gravity  is 

weight  of  a  hody  x^^ 

Specific  gravity  =  -  ,  J  .        —  (4) 

weight  of  an  equal  vol.  of  water 

127.  Density  of  Liquids. — There  are  several  methods 
for  finding  the  density  of  a  liquid.     Let  a  solid  that  is 
denser  than  water  be  weighed  in  air,  then  in  water,  and 
finally  in  a  liquid  whose  density  is  to  be  found.     The  loss  of 
weight  in  water  then  represents  numerically  the  volume 
of  the  submerged  solid.    The  loss  of  weight  in  the  liquid  of 
unknown  density  represents  the  mass  of  an  equal  volume 
of  that  liquid.    Therefore  the  density  of  the  liquid  is  found 
by  dividing  the  latter  loss  by  the  former.     For  example,  a 
solid  weighs  80  grams  in  air,  55  grams  in  water,  and  60 
grams  in  another  liquid.     The  mass  of  water  displaced  is 
25  grams  and  that  of  the  other  liquid  20  grams.     Hence 
the  volume  of  the  liquid  of  unknown  density  displaced  by 
the  solid  is  25  cubic  centimeters,  and  its  density  20  -+•  25, 
or  0.8  gram  per  cubic  centimeter. 

128.  Hydrometers.  —  The  law  of  floating  bodies  (§  122) 
affords  a  convenient  method  for  finding  the  density  of  a 
liquid.     Let  the  bar  of  wood  used  in  Experiment  1,  §  122, 
be  placed  in  a  jar  of  water  and  the  volume  of  the  sub- 
merged  portion   ascertained.     Next   place   the  bar  in  a 
liquid  whose  density  is  to  be  found,  and  again  find  the 
volume  of  the  part  beneath  the  liquid.    If  the  displacement 
of  water  is  v1  cubic  centimeters,  the  weight  of  the  displaced 
water  is  v^  grams.     If  d  is  the  density  of  the  other  liquid 
and  vz  its  displacement,  the  weight  of  the  liquid  displaced 
is  v2d  grams.     According  to  the  law  of  floating  bodies  the 
weight  of  the  liquid  displaced  in  each  instance  is  equal  to 
the  weight  of  the  floating  bar.     Hence  the  two  displace- 
ments are  of  equal  weight.     We  may  therefore  write 

v2d  =  vx ;  whence  d  =  — -•  (5) 


128 


A  HIGH   SCHOOL  COURSE  IN   PHYSICS 


FIG.    93.— 


Hydrometer 
of  Constant 
Weight. 


A  floating  body  that  is  made  in  a  convenient  form  for 
measuring  the  densities  of  liquids  is  called  an  hydrometer. 
If  the  weight  of  the  hydrometer  remains  con- 
stant as  in  the  case  of  the  wooden  bar  used 
in  the  experiment  just  described,  the  instru- 
ment is  called  an  hydrometer  of  constant 
weight.  In  some  instances  the  densities  are 
indicated  upon  the  bar  and  may  be  read  off 
at  once  by  observing  the  point  to  which  the 
instrument  sinks.  Many  forms  of  hydrom- 
eters are  in  daily  use,  the  size,  shape,  and 
graduations  being  adapted  to  the  particular 
commercial  purpose  that  they  serve.  A  com- 
mon form  is  shown  in  Fig.  93.  A  cylindri- 
cal glass  tube  terminates  below  in  a  small 
bulb  filled  with  mercury,  which  causes  the 
instrument  to  float  in  an  upright  position.  The  upper  por- 
tion of  the  tube  is  made  small,  and  the  graduations  are 
upon  a  paper  scale  sealed  within. 

Nicholson's  Hydrometer.  —  The  Nicholson  hydrometer  shown  in 
Fig.  94  represents  the  type  known  as  hydrome- 
ters of  constant  volume,  i.e.  of  constant  displace- 
ment. This  instrument  is  used  in  finding  the 
density  of  a  solid  body.  Let  wl  grams  be  the 
weights  required  in  the  upper  pan  A  ,  to  sink  the 
hydrometer  to  a  certain  mark  placed  on  the  slen- 
der stem  a.  After  placing  the  solid  of  unknown 
density  in  the  upper  pan,  the  weight  required  to 
sink  the  instrument  to  the  same  mark  is  w2  grams. 
Then  wl  —  w2  is  the  mass  of  the  solid.  Let  ws  be 
the  weights  required  in  the  upper  pan  after  the 
solid  has  been  transferred  to  the  lower  pan  R. 
Then  w3  —  w2  is  the  mass  of  the  water  displaced 
and  numerically  equal  to  the  volume  of  the  solid. 

Therefore,  the  numerical  value  of  the  density 
of  the  solid  expressed  in  metric  units  is  : 


Constant     Volume 
Hydrometer. 


MECHANICS   OF   LIQUIDS  129 


Density  =      -  (6) 

Wo    —    Wo 


EXERCISES 

1.  A  piece  of  lead  weighs  56.75  g.  in  air  and  51.73  g.  when  sus- 
pended in  water.     Find  the  volume  and  density  of  the  lead. 

2.  A  cylinder  of  aluminium  weighs  28.35  g.  in  air  and  17.85  g. 
when  immersed  in  water.     Calculate  the  volume  and  density  of  the 
metal.     Compute  the  sp.  gr.  of  aluminium. 

3.  A  piece  of  glass  weighing  45  g.  in  air  weighs  22.5  g.  in  water 
and  23.75  g.  in  oil.     Calculate  the  densities  of  the  glass  and  the  oil. 

4.  What  would  be  the  weight  of  the  glass  in  Exer.  3  when  im- 
mersed in  a  liquid  whose  density  is  0.922  g.  per  cubic  centimeter? 

5.  The  density  of  marble  is  2.7  g.  per  cubic  centimeter.    What  is 
the  weight  of  a  rectangular  block  1  m.  long,  40  cm.  wide,  and  15  cm. 
thick  ?     Compute  the  sp.  gr.  of  marble.    What  is  its  mass  per  cu.  ft.  ? 

6.  Silver  is  10.4  times  as  heavy  as  an  equal  volume  of  water. 
What  will  20  g.  of  silver  weigh  when  immersed  in  water? 

7.  Ice  is  0.9  as  heavy  as  an  equal  volume  of  water.     If  a  piece  of 
ice  weighing  500  g.  floats  on  water,  what  is  the  volume  of  the  sub- 
merged portion  ?    What  is  the  volume  of  the  ice? 

SUGGESTION.  —  Apply  the  law  of  floating  bodies  and  ascertain  the 
weight  of  water  displaced. 

8.  A  bar  of  wood  weighing  100  g.  floats  on  water  with  0.82  ©f  its 
volume  submerged;  when  placed  in  oil,  0.80  of  its  volume  is  submerged. 
Calculate  the  sp.  gr.  and  density  of  the  oil. 

9.  A  piece  of  paraffin  weighs  69  g.  in  air  and  when  attached  to  a 
sinker  and  suspended  in  water,  85.8  g.     If  the  weight  of  the  sinker  in 
water  is  95.7  g.,  what  is  the  volume  and  density  of  the  paraffin? 

10.  The  weight  required  to  sink  a  Nicholson  hydrometer  to  the 
mark  on  the  stem  is  45  g. ;  when  a  piece  of  marble  is  placed  in  the 
upper  pan,  the  weight  required  is  15.3  g.  and  with  the  marble  in 
the  submerged  pan  26.3  g.     Find  the  density  of  the  marble. 

11.  Find  the  density  of  paraffin  from  the  following  data : 
Weight  required  to  sink  Nicholson's  hydrometer  to  mark  56.4  g. 
Weight  to  sink  hydrometer  with  paraffin  in  upper  pan     45.6  g. 
Weight  required  with  paraffin  in  submerged  pan  57.6  g. 

12.  The  density  of  mercury  is  13.59  g.  per  cubic  centimeter.     If  a 
cubic  foot  of  water  weighs  62.5  lb.,  what  is  the  weight  of  a  cubic  inch 
of  mercury? 

10 


130          A  HIGH   SCHOOL  COURSE  IN   PHYSICS 


5.    MOLECULAR   FORCES   IN   LIQUIDS 

129.  Cohesion  and  Adhesion.  —  Many  of  the  phenomena 
of  nature  lead  to  the  conclusion  that  bodies  are  made  up 
of  extremely  minute  particles  to  which  is  given  the  name 
molecules.      The  molecules  of  bodies  possess  more  or  less 
freedom  of  motion  among  themselves  depending  on  the 
nature  of  the  body ;  this  freedom  is  greatest  in  gases  and 
least  in  solids.     It  is  due  to  the  very  perfect  freedom  of 
motion  among  the  molecules  of  a  liquid  (i.e.  to  the  prop- 
erty of  fluidity)  that  they  are  able  to  transmit  pressures 
in  all  directions,  thus  giving  rise  to  the  principle  stated  in 
(1)  §  104  and  to  Pascal's  Law  discussed  in  §  110. 

When  near  together  molecules  attract  one  another,  thus 
producing  the  resistance  found  when  we  attempt  to  break 
a  wire,  to  remove  paint  from  glass,  to  tear  paper,  etc. 
The  name  cohesion  is  given  to  this  attraction  when  it  is 
between  molecules  of  the  same  kind,  and  the  term  adhesion 
applies  to  the  attraction  when  the  molecules  are  of  differ- 
ent kinds. 

Cohesion  serves  to  bind  individual  molecules  into  bodies, 
or  masses,  and  adhesion  to  hold  together  bodies  of  differ- 
ent kinds.  Two  clean  surfaces  of  lead  will  cohere  when 
pressed  firmly  together,  and  the  dentist  hammers  gold  leaf 
into  a  solid  lump  to  form  the  filling  for  a  tooth.  The 
blacksmith  brings  together  white-hot  pieces  of  iron  and 
by  blows  brings  their  molecules  into  such  close  proximity 
that  cohesion  takes  place.  Clean  graphite  powder  is 
pressed  into  a  solid  mass  to  form  the  lead  of  a  pencil ;  the 
force  of  cohesion  holds  the  particles  together.  The  attrac- 
tion between  wood  and  glue,  stone  and  cement,  paint  and 
iron,  etc.,  affords  cases  illustrating  the  force  of  adhesion. 

130.  Surface   Films   of    Liquids.  —  Although    steel    is 
nearly  eight   times   as   dense   as  water,  a  small   sewing 


MECHANICS   OF  LIQUIDS  131 

needle  may  be  caused  to  "  float "  on  water  by  placing  it 
carefully  upon  the  surface.  If  the  surface  of  the  liquid 
is  closely  examined,  it  will  be  observed  that  the  needle 

rests  in  a  slight  depression,  as  shown  |g^ _^ 

in  Fig.   95.     In  fact,  the  appearance 
is  as  if  a  thin  membrane  were  stretched    ~ 

FIG.    95.  —  An  Ordinary 

across  the  surface  of  the  water.  steel  Needle  "Floating" 

The   three    following    experiments  on  Water> 

show  an  important  property  of  the  surface  films  of 
liquids : 

1.  Let  a  soap  bubble  be  blown  on  the  bowl  of  a  clay  pipe  or  the 
mouth  of  a  small  glass  funnel.     If  the  stem  is  left  open  a  moment,  it 
may  be  observed  that  the  bubble  diminishes  in  size.     A  candle  flame 
held  near  the  opening,  in  the  stem  will  be  deflected  by  the  current  of 
air  that  is  forced  out  by  the  contracting  bubble. 

2.  Let  a  loop  of  thread  be  tied  to  one  side  of  a  wire  frame  4  or  5 
centimeters  in  diameter  so  that  it  will  hang  near  the  center,  as  shown 

in  (1),  Fig.  96.  If  now  the  frame 
is  dipped  into  a  soap  solution,  a 
liquid  film  will  form  across  it  with 
the  loop  closed,  as  shown.  If,  how- 
ever, the  film  within  the  loop  of 
thread  be  broken  by  means  of  a  hot 
wire,  the  loop  instantly  opens  out 
into  a  circle,  as  shown  in  (2). 

3.   Let  a  mixture  be  made  of  alco- 
*2\  hoi  and  water  of  such  strength  that 

FIG.  96.  — An  Effect  of  Surf  ace  Ten-  a  drop  of  olive  oil  will  remain  in  it 
sion  in  a  Liquid  Film.  at  any  depth.     The  oil  may  be  in- 

troduced beneath  the  surface  by  means  of  a  small  glass  tube.  It 
will  be  found  that  the  globule  of  oil  at  once  assumes  a  spherical 
form.  In  order  to  avoid  an  apparent  flattening  of  the  drop  of  oil,  a 
body  with  flat  sides  should  be  used. 

These  experiments  show  that  the  surface  film  of  a  liquid 
tends  to  contract  and  become  as  small  as  possible.  In 
Experiment  1  the  contraction  produces  a  pressure  that 
drives  the  air  from  the  tube.  In  Experiment  2  the  film 


132          A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

assumes  the  least  possible  area.  This  is  the  case  when 
the  area  within  the  loop  is  as  large  as  it  can  become,  i.e. 
when  it  is  circular.  In  Experiment  3  the  film  of  oil  com- 
pletely incloses  the  globule  and  causes  the  oil  to  assume 
the  geometrical  form  requiring  the  least  superficial  area 
for  a  given  volume.  The  spherical  form  fulfills  this  con- 
dition. For  a  similar  B  reason  a  soap  bubble  takes  the 
shape  of  a  sphere. 

131.  Surface  Tension.  —  It  will  be  seen  from  a  study  of 
Fig.  97  that  the  conditions  under  which  the^surface  mole- 
cules of  a  liquid  exist  is  very 
different' from  that  of  the  mole- 
cules lower  down.  Molecule 
A  at  the  center  of  the  small 
circle  is  attracted  equally  in  all 


FIG.  97.— Molecular  Forces  near  directions  by  the  neighboring 
the  Surface  of  a  Liquid.  molecules.  This  is  the  force  of 
cohesion  and  acts  across  a  very  small  distance  represented 
here  by  the  radius  of  the  circle.  Very  near  the  surface  the 
forces  acting  downward  on  molecule  B  are  greater  than 
those  that  act  upward,  while  for  molecule  O  there  is  no 
upward  attraction  whatever.  Hence  the  surface  layers 
of  molecules  are  greatly  condensed  by  this  excess  of  force 
acting  always  toward  the  body  of  the  liquid.  In  this  man- 
ner is  formed  a  tough,  tense  surface  film  whose  constitution 
is  different  from  that  in  the  interior  mass.  The  measure 
of  the  tendency  of  the  surface  layers  of  a  liquid  to  contract 
is  called  surface  tension. 

132.  Capillary  Phenomena.  —  The  effects  of  surface 
films  were  first  investigated  in  small  glass  tubes  of  hair- 
like  dimensions.  Hence  the  name  "  capillarity "  arises 
from  the  Latin  word  capillus,  meaning  "hair."  Figure 
98  represents  a  series  of  glass  tubes  varying  from  0.2  mil- 
limeter to  2  millimeters  in  diameter.  If  the  tubes  are 


MECHANICS   OF  LIQUIDS 


133 


moistened  and  then  set  upright  in  a  shallow  vessel  of 
water,  the  liquid  will  rise  in  the  tubes,  —  highest  in  the 
smallest  tube  and  least  in  the  largest.  It 
may  also  be  observed  that  the  surface  of  the 
liquid  turns  upward  wherever  it  comes  in 
contact  with  the  glass.  Hence  the  surface 
within  the  tube  is  concave,  and  a  so-called 
meniscus  is  formed. 

If,  now,  glass   tubes   of   small   bore   are 
placed,  in  mercury  (Fig.    99),    the  -liquid, 

which  in  this  case  does  FlG;.  98--Eleva: 

tion  of  a  Liquid 
not  Wet  the  glass,  will        in  Capillary 

be  depressed  within 
the  tubes.  The  depression  is  great- 
est in  the  smallest  tube  and  least  in 
the  largest.  The  edges  of  the  film 
in  contact  with  the  glass  may  be 
seen  to  turn  downward  so  that  the 
FIG.  99.  —  Depression  of  surfaces  within  the  tubes  are  convex, 

Mercury    in     Capillary 

Tubes.  and  the  meniscus  thus  formed  is  111- 

^/_  verted. 

The  following  laws  of  capillary  action  may  be  stated : 

(1)  Liquids  are  elevated  in  tubes  which  they  wet,  but  are 
depressed  in  tubes  which  they  do  not  wet. 

(2)  The  elevation  or  depression  is  inversely  proportional 
to  the  diameter  of  the 

tubes.  ^  K  D 


133.    Capillary  Ac- 
tion   Explained.  —  If 

a  plate  of  glass,  shown 
in  cross  section  in 
(1),  Fig.  100,  be 
placed  upright  in  a 


(1)  (Z) 

FIG.  100.  —  Water  Lifted  by  the  Contraction  of 
the  Surface  Film. 


134          A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

shallow  vessel  of  water  after  having  been  moistened, 
there  will  be  formed  a  continuous  film  ABO  lying  partly 
upon  the  glass  and  partly  upon  the  water.  On  account 
of  the  tendency  of  this  film  to  contract,  as  shown  in  §  131, 
the  corner  at  B  will  be  rounded  and  a  small  portion  of  the 
water  lifted  against  the  glass. 

Again,  if  a  moistened  tube  of  glass,  (2),  Fig.  100,  be 
dipped  into  water,  a  film  A  B  CD  is  formed  adhering  to  the 
glass  and  extending  across  the  water  in  the  tube.  Ow- 
ing to  the  tendency  of  this  film  to  contract,  a  force  is  pro- 
duced that  is  sufficient  to  lift  the  column  of  water  BEFC 
in  opposition  to  the  force  of  gravity.  When  the  tube  is 
dipped  into  mercury,  the  liquid  forms  no  film  adhering  to 
the  glass  above  the  surface  level.  On  the  other  hand,  the 
surface  film  of  mercury  (see  Fig.  99)  continues  down- 
ward along  the  glass  walls  of  the  tube  and  then  upward 
into  the  tube  at  the  lower  end.  The  force  which  is 
developed  by  the  contraction  of  this  film  tends  to  pull 
the  liquid  down  within  the  tube.  The  depression  thus 
produced  is  such  that  the  downward  force  of  the  con- 
tracting film  is  balanced  by  the  upward  pressure  of  the 
liquid  within  the  tube. 

134.  Capillary  Action  in  Soils.  —  The  principles  of 
capillary  action  find  an  important  application  in  the 
distribution  of  moisture  in  the  soil.  In  compact  soil 
water  is  brought  to  the  surface  in  much  the  same  manner 
as  it  rises  in  a  piece  of  loaf  sugar  which  is  allowed  to 
come  in  contact  with  it.  As  rapidly  as  evaporation  goes 
on  at  the  surface,  the  loss  is  supplied  by  capillarity  from 
below.  In  dry  weather  it  is  desirable  to  prevent  this 
surface  loss,  which  is  done  by  "  mulching  "  and  loosen- 
ing the  soil  by  cultivation.  In  this  latter  process  the 
grains  of  soil  are  broken  apart  and  the  interstices  thus 
made  too  large  for  effective  capillary  action  to  take  place. 


MECHANICS   OF   LIQUIDS  135 

The  moisture  then  rises  to  a  level  a  few  inches  beneath 
the  surface,  where  it  is  made  use  of  by  growing  plants. 

EXERCISES 

1.  Why  is  a  drop  of  dew  spherical?    Examine  a  small  globule  of 
mercury  placed  on  glass.     Ho'w  do  you  account  for  its  form  ? 

2.  By  heating  a  piece  of  glass  until  it  softens  the  sharp  corners 
become  rounded  and  smooth.     Explain. 

3.  Tn   the  manufacture  of  shot  molten  lead  is  poured  through 
a  small  orifice  at  the  top  of  a  tower.     In  falling  the  stream  breaks 
up  into  drops  which  solidify  before  reaching  the  earth.     What  gives 
the  spherical  form  to  these  masses  of  lead  ? 

4.  Why  does   oil  flow  upward  in  the   wick  of  a  lamp?     Will 
mercury  do  the  same? 

5.  Grease  may  be  removed  from  a  piece  of  cloth  by  covering  it 
with  blotting  paper  and  passing  a  hot  flatiron  over  it.     Explain. 

6.  Place  two  toothpicks  upon  water  about  a  centimeter  apart. 
Touch  the  liquid  surface  between  them  with  a  glass  rod  moistened 
with  alcohol.     From  the  manner  in  which  the  pieces  of  wood  move 
about,  observe  which  liquid  film  has  the  greater  tension. 

7.  Explain  how  an  insect  can  run  on  the  surface  of  water  without 
sinking. 

8.  Explain  the  action  of  a  towel ;  of  blotting  paper ;  of  sponges. 

9.  Why  can  we  not  write  with  ink  upon  unglazed  paper? 

10.  Why  are  the  footprints  made  in  newly  cultivated  soil  moist 
while  the  loose  earth  is  dry? 

11.  Explain  why  it  is  necessary  to  pack  the  earth  around  plants 
and  small  trees  when  they  are  first  set  out.     Later  on  we  loosen  the 
soil  about  them.     Why? 

SUMMARY 

1.  The  force  exerted  by  a  liquid  of  uniform  density 
against  a  given   surface   is  directly  proportional  to   the 
depth  of  the  surface  (§  103). 

2.  At  a  given  depth  the  force  exerted  by  a  liquid  is  the 
same  in  all  directions  and  acts  always  in  a  direction  per- 
pendicular to  the  surface  against  which  it  presses  (§  104). 

3.  The  force  exerted  by  a  liquid  on  any  horizontal  sur- 
face is  equal  to  the  product  of  the  area  and,  depth  of  the 


136          A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

surface  and  the  density  of  the  liquid.     The  equation  is  *• 
total  pressure  =  ahd  (§  106). 

4.  When  the  surface  pressed  against  is  not  horizontal, 
the  product  of  the  area  and  the  density  must  be  multiplied 
by   the   depth   of  the   center  of  mass  of  the  given  sur- 
face (§  106). 

5.  The  force  exerted  by  a  liquid  against  the  bottom  of 
a  vessel  of  given  depth  is  independent  of  the  form  of  the 
vessel  (§  107). 

6.  A  liquid  at  rest  in  communicating  vessels  remains 
at  the  same  level  in  all  its  parts  (§  109). 

7.  A  force  applied  to  any  area  of  a  confined  liquid  is 
transmitted  undiminished  by  the  liquid  to  every  equal 
area  of  the  interior  of  the  containing  vessel  and  to  every 
part  of  the  vessel.     This  is  known  as  Pascal's  Law  (§  110). 

8.  The  mechanical  advantage  of  the  hydraulic  press  is 
the  ratio  of  the  area  of  the  large  piston  to  that  of  the  small 
one  (§  111). 

9.  A  body  immersed  in  a  liquid  is  buoyed  up  by  a 
force  equal  to  the  weight  of  the  liquid  that  it  displaces. 
This  is  known  as  Archimedes'  Principle  (§  120). 

10.  A  floating  body  displaces  a  mass  of  the  liquid  in 
which  it  floats,  whose  weight  equals  its  own  (§  122). 

11.  The  volume  of  a  body  insoluble  in  water  may  be 
found,  according  to  Archimedes'  Principle,  by  ascertaining 
the  weight  of  water  that  it  will  displace  (§  123). 

12.  Density  =  maSS     (§  125). 

volume 

13.  Bodies  are  assumed  to  be  composed  of  extremely 
small   particles    called  molecules.     When  near  together, 
molecules  attract  each  other.      Cohesion  is  the  attraction 
between  molecules  of  the  same  kind ;  adhesion  is  the  at- 
traction between  molecules  of  different  kinds  (§  129). 


MECHANICS   OF   LIQUIDS  137 

14.  The  surface  of  a  liquid  tends  to  contract  and  be- 
come as  small  as  possible;   hence  the  spherical  form  as- 
sumed by  soap  bubbles,  dew  drops,  globules  of  mercury, 
etc  (§130). 

15.  Liquids  are  elevated  in  tubes  which  they  wet,  but 
are  depressed  in  those  which  they  do  not  wet.     The  ele- 
vation or  the  depression  is  inversely  proportional  to  the 
diameter  of  the  tube  (§  132). 

16.  Moisture  rises  readily  in  compact  soils,  but  ceases 
to  rise  in  those  of  loose  texture.     Hence  the  loss  of  soil 
water  by  evaporation  is  effectually  prevented  by  "mulch- 
ing "  or  by  cultivation  (§  134). 


CHAPTER   VIII 

MECHANICS  OF   GASES 

1.   PROPERTIES   OF   GASES 

135.  Characteristics    of    Gases.  —  Gases,   like    liquids, 
possess  the  property  of  fluidity,  i.e.  they  may  be  deformed 
by  any  force  however  small.     But  a  gas  differs  from  a 
liquid  in  that  it  has  no  definite  size  of  its  own  ;  it  not 
only  fits  itself  to  the  shape  of  the  vessel  containing  it, 
but  always  entirely  fills  it.     On  account  of  their  common 
property  of  fluidity,  liquids  and  gases  are  classed  together 
as  fluids.     As  a  consequence  of  this  property,  the  laws  of 
pressure  relative  to  liquids  stated  in  1,  §  104  and  in  §  110 
are  equally  applicable   to  gases.      Other  laws,  however, 
arise  in  the  case  of  a  gas  on  account  of  the  tendency  to 
adapt  its  size  to  the  capacity  of  the  containing  vessel. 

136.  Laws  Common   to  Liquids  and  Gases.  —  (1)  The 
force    exerted    against    any  surface   is   perpendicular   to 
that  surface.     (2)  The  force  at  any  point  is  the  same  in 
all  directions  (§  104).      (3)  An  immersed  body  is  buoyed 
up  by  a  force  equal  to  the  weight  of  the  liquid  or  the  gas 
that  it  displaces  (§  120). 

137.  Weight  and  Density  of  Air.  —  We  are  taught  by 
everyday  experience  that  a   gaseous  medium  which  we 
call  air  surrounds  us  on  every  hand.     The  bubbles  that 
may  be  produced  by  blowing  through  a  tube  inserted  into 
water,  the  process  of  breathing,  the  resistance  that  the  air 
offers  to  a  rapidly  moving  bicycle  or  train,  the  various 
effects  of  the  wind,  and  many  other  phenomena  demon- 
strate to  us  the  presence  of  this  medium. 

138 


MECHANICS   OF  GASES  139 

It  was  shown  by  experiment  in  §  3  that  air  has  weight. 
The  same  apparatus  may  be  used  in  finding  the  density  of 
air  in  the  following  manner: 

Let  the  bulb  be  weighed  carefully  before  and  after  admitting  the 
air.  The  increase  in  weight  will  give  the  mass  of  the  air  contained  in 
the  bulb.  The  capacity  of  the  bulb  is  next  ascertained  by  filling  it 
with  water  and  weighing  it.  Dividing  the  mass  of  the  air  by  the 
capacity  of  the  bulb  gives  the  density  of  the  air. 

The  density  of  air  at  the  temperature  of  freezing  water 
and  under  the  average  sea-level  pressure  is  0.001293  gram 
per  cubic  centimeter.  Hence  a  liter  (1000  cm.3)  of  air 
weighs  nearly  1..3  grams.  A  cubic  foot  of  air  weighs  about 
an  ounce  and  a  quarter  ;  hence  12  cubic  feet  weigh  nearly 
a  pound.  The  amount  of  air  in  an  ordinary  schoolroom 
weighs  more  than  half  a  ton.  Thus  we  live  submerged 
in  an  ocean  of  air  which  extends  many  miles  above  us  and 
exerts  a  pressure  of  nearly  15  pounds  per  square  inch. 

2.     PRESSURE  OF  THE  AIR  AGAINST  SURFACES 

138.  Atmospheric  Pressure.  —  Since  air  has  weight  and 
fluidity,  the  atmosphere  must  exert  a  force  against  all 
surfaces  with  which  it  comes  in  contact.  The  existence  of 
such  a  force  may  be  shown  by  the  following  experiments  : 

1.  Fill  a  tumbler  with  water  and  invert  it  in  a  vessel  of  water. 
Lift  the  inverted  tumbler  until  its  opening  is  horizontal  and  just 
submerged,  as  in  Fig.  101.     The  water  re- 

mains in  the  tumbler  because  it  is  sup- 
ported by  the  force  exerted  by  the  atmos- 
phere downward  against  the  free  surface 
of  water  in  the  vessel. 

2.  While  the  inverted  tumbler  is  in  the 
condition  shown  in  Fig.  101,  place  a  piece 
of  cardboard  across  its  mouth,  pressing  it 
close  against  the  rim.     Carefully  lift  the 
tumbler  from  the  water,  and  the  cardboard 


will  not   fall  off.     The   force   exerted   by         Water. 


140 


A   HIGH   SCHOOL   COURSE   IN   PHYSICS 


the  air  against  the  cardboard  in  an  upward  direction  is  sufficient  to 
support  the  weight  of  the  water  in  the  tumbler,  as  in  Fig.  102. 

3.    Tie  a  piece  of  sheet  rubber  over  a 

glass  vessel,  as  shown  in  Fig.  103.     Place 

the  vessel  on  an  air  pump    and  exhaust 

the  air  from  the 

space       beneath 

the  rubber.    The 

membrane,      no 

longer  supported 

by  the  air  from 

below,    is    more 

and     more     de- 
pressed  until   it 


FIG.  102.  —Action  of  Atmos- 
pheric Pressure  Upward. 


FIG.  103.  —  Atmospheric 
Pressure  upon  a  Rubber 
Membrane. 


finally  bursts  under  the  pressure  of  the  air 
above. 

4.   If  a  glass  tube  3  or  4  feet  long  (or 
even  longer)  can  be  secured,  place  it  nearly 

vertical  with  one  end  in  a  tumbler  of  water.  Try  to  elevate  the 
water  to  the  top  of  the  tube  by  "sucking"  on  the  upper  end  of 
it.  Now  insert  the  lower  end  of  the  tube  in  mercury  and  try  to 
elevate  it  in  the  same  manner.  It  is  so  much  denser  than  water, 
that  it  can  be  lifted  only  a  few  inches. 

139.  Limitations  of  Atmospheric  Pressure. — The  atten- 
tion, of  Galileo  was  called  to  the  fact  that  in  wells  of 
unusual  depth  suction  pumps  (§  155)  were  unable  to  lift 
water  more  than  about  32  feet  above  the  level  of  the  water 
in  the  wells.  At  that  time  it  was  supposed  that  "  nature 
abhorred  a  vacuum  "  and  that  she  hastened  to  fill  all  such 
spaces  with  whatever  material  happened  to  be  most  avail- 
able. Thus  Galileo  was  led  to  believe  that  nature's  abhor- 
rence for  a  vacuum  had  its  limitations,  and  he  probably 
suspected  that  the  rise  of  water  in  pumps  was  due  to*  air 
pressure  on  the  surface  of  the  water.  It  remained  for  his 
pupil,  Torricelli,  however,  to  devise  a  suitable  method  for 
measuring  the  actual  pressure  of  the  atmosphere.  He  suc- 
ceeded in  doing  this  in  the  year  1643. 


MECHANICS   OF  GASES 


141 


FIG.  104. 


140.  Torricelli's  Experiment.  —  This  important  histori- 
cal experiment  is  performed  by  making  use  of  a  strong 
glass  tube  about  80  centimeters  long  which  is  closed  at  one 
end.     The  tube  is  first  filled  with 

mercury  in  order  to  expel  the  air, 
after  which  it  is  closed  by  the 
finger  and  inverted  in  a  vessel  of 
mercury,  Fig.  104,  care  being 
taken  to  prevent  the  entrance  of 
air.  The  mercury  falls  at  once 
and  leaves  a  vacuum  of  several 
centimeters  at  the  top  of'  the 
tube.  The  force  due  to  the  at- 
mosphere which  is  exerted  down- 
ward against  the  free  surface  of 
mercury  in  the  vessel  is  trans- 
mitted to  the  interior  of  the  tube 
in  which  it  is  able  to  support 
a  column  of  the  liquid  usually  about  75  centimeters  in 
height. 

141.  Pascal's  Experiment.  —  Pascal  reasoned  that  if  the 
column  of  mercury  in  a  Torricellian  tube  were   indeed 
supported  by  the  atmosphere,  the  column  should  become 
shorter  at  a  high  altitude.     The  column  was  accordingly 
measured  at  the  top  of  a  high  tower  in  Paris  and  a  small 
decrease  detected.     Five  years  after  Torricelli's  discovery, 
the  tube  was  carried  to  the  top  of  the  Puy  de  Dome,  a 
high  mountain  in  Auvergne,  France,  where  a  test  showed 
a  decided  decrease  of  about  8  centimeters  in  the  height  of 
the  mercurial  column. 

142.  Variations  in  Atmospheric  Pressure.  —  The  atmos- 
pheric pressure  at  a  given  place  is  far  from  being  constant. 
The  variations  at  any  given  place  amount  to  about  3  cen- 
timeters.    At  sea-level  the  average  height  of  the  mercurial 


Torricelli's  Experi- 
ment. 


142         A  HIGH   SCHOOL  COURSE  IN   PHYSICS 


column  is  about  76  centimeters ;  hence  this  height  is  taken 
as  the  standard  of  pressure  and  is  called  mean  sea-level 
pressure.  Any  pressure  equivalent  to  a  column  of  mer- 
cury 76  centimeters  high  is  said  to  be  "  one  atmosphere.." 
143.  The  Barometer.  —  The  mercurial  barometer  (pro- 
nounced ba  rom'e  ter)  is  merely  a  mounted  Torricellian 
tube  for  showing  the  pressure  of  the  atmos- 
phere. In  order  that  it  may  give  correct 
indications,  the  mercury  must  be  pure,  and 
clean,  and  the  space  above  the  liquid  in  the 
tube  as  free  as  possible  from  the  presence  of 
air  and  other  gases;  i.e.  the  vac- 
uum must  be  as  nearly  perfect 
as  possible.  Atmospheric  pres- 
sure is  usually  measured  in  inches 
or  centimeters  of  mercury.  The 
space  above  the  mercury  in  the 
tube  is  known  as  the  Torricellian 
vacuum. 

As  the  height  of  the  mercurial 
column  changes  with  the  pressure 
of  the  air,  the  surface  of  the 
liquid  rises  and  falls  in  the  res- 
ervoir, or  cistern,  below.  For  an 
accurate  reading  of  the  barometer 
the  mercury  column  must  be  meas- 
ured from  the  surface  of  the  liquid 
in  the  reservoir..  In  the  barome- 
ter shown  in  Fig.  105  the  reser- 
voir has  a  flexible  bottom.  By 
FIG.  105.  -  A  standard  Mer-  turninff  the  screw  S  the  surface 

curial  Barometer. 

of  the  mercury  in  the  cistern  is 

brought  to  the  zero  point  of  the  scale,  whose  position  is 
marked  by  an  ivory  index  B.     The  height  of  the  mercury 


MECHANICS   OF  GASES 


143 


is  then  read  by  observing  the  position  of  the 
upper  surface  of  the  liquid  in  the  tube  at  A. 

Some  mercurial  barometers  have  the  form 
shown  in  Fig.  106.  The  height  of  the  mer- 
cury is  found  by  reading  the  positions  of  the 
two  liquid  surfaces  A.  and  .Z?,  in  which  case 
the  difference  between  the  two  readings  gives 
the  length  of  the  mercury  column  supported 
by  the  atmosphere. 

144.  The  Aneroid  Barometer.  —  A  barometer  of 
the  form  shown  in  Fig.  107  is  in  common  use.  As  the 
name  indicates,  the  aneroid  barometer  is  "without 
liquid,"  and  depends  for  its  operation  on  a  circular 
chamber  C 
made  of  thin 
metal  with 
corrugated 
sides.  The 
air  in  the 
chamber  is 
partly  re- 
moved, after 
which  the  chamber  is  hermetically  sealed.  An  increase 
of  atmospheric  pressure  forces  the  side  of  the  box 
inward,  but  a  decrease  allows  it  to  spring  out. 
This  motion, 
Chough  very 
s^ght,  is 
transmitted 


FIG.  107.  —  The  Aneroid  Barometer. 


Fxo.106.-A 

Mercurial 

Barometer, 

through    the 

multiplying  systems  of  levers 
L  and  A  and  the  small  chain 
B  to  the  axle  S  which  car- 
ries the  index,  or  movable 
pointer,  /.  The  scale  is 
graduated  to  correspond  to 

the   readings  of   a  standard     FIG;  io8._The  Barograph,  or  Self-registering 
mercurial  barometer.    These  Barometer. 


144 


A  HIGH   SCHOOL  COURSE  IN   PHYSICS 


instruments  are  made  of  convenient  size  to  be  used  by  surveyors  and 
explorers  in  ascertaining  altitudes.  It  should  be  said  that  the  words 
"  Rain,"  "  Fair,"  etc.,  printed  on  the  dial  of  an  aneroid  barometer  are 
merely  indications  of  the  general  trend  of  weather  changes. 

The  principle  of  the  aneroid  is  employed  in  the  "  barograph,"  or 
self-recording  .barometer,  shown  in  Fig.  108.  This  instrument  is 
provided  with  an  index  which  carries  a  pen  that  makes  a  continuous 
record  of  the  atmospheric  pressure  on  a  revolving  cylinder  covered  with 
suitable  paper.  A  portion  of  such  a  record  is  shown  in  Fig.  109. 


THURSDAY          / 


FRIDAY 


SATURDAY 

4   6  8  16 jf'2   4  6   I 


JNDAY  / 

p^'2   ji    6  8  1,6yT" 


J  8  10^JJ_8rbSrt.2_4  6 


M    8 


1tOYu2    468  1,0 


FIG.  109.  —  Portion  of  a  Record  Made  by  a  Barograph. 

145.  Utility  of  the  Barometer. — The  barometer  is  an 
important  laboratory  instrument,  inasmuch  as  the  atmos- 
pheric pressure  must  be  known  in  carrying  out  many 
experiments  in  both  Physics  and  Chemistry.  In  this 
respect  it  ranks  in  usefulness  with  the  thermometer, 
balance,  etc. 

From  the  readings  of  barometers  taken  simultaneously 
at  many  places  of  observation  and  telegraphed  to  central 
stations,  the  direction  of  atmospheric  movements  can  be 
predicted.  Thus  the  barometer  becomes  an  aid  in  fore- 
casting the  weather.  Furthermore,  a  "low"  barometer, 
i.e.  decreased  pressure,  usually  accompanies  or  precedes 


MECHANICS   OF  GASES 


145 


.29.7 


stormy  weather,  while  a  rising  barometer  generally  de- 
notes the  approach  of  fair  weather.  If  a  weather  map 
be  consulted,  certain  regions  will  be  found  marked 
"High"  and  others  marked  "Low."  The  places  at  which 
the  pressures  are  equal  are  joined  by  curves  called 
isobars,  upon  each  of  which  is  indicated  the  barometric 
reading,  Fig.  110. 
The  direction  of  the 
wind  at  each  place  of 
observation  is  indi- 
cated by  an  arrow. 
The  general  direc- 
tion of  the  wind  is  al- 
ways from  places  of 
"high"  toward  those 
of  "  low  "  pressure. 

Another  important 
use  of  the  barometer 
is  made  in  measuring 
the  difference  in  alti- 
tude of  two  places. 
This  measurement  de- 
pends upon  the  fact 
that  the  atmospheric 
pressure  decreases  with  the  elevation  above  sea-level. 
For  places  not  far  above  the  level  of  the  sea  the  decrease 
is  about  1  millimeter  for  every  10.8  meters  of  elevation, 
or  0.1  inch  for  every  90  feet  of  ascent.  The  decrease  in 
pressure  as  one  climbs  a  mountain  is  easily  accounted  for 
when  it  is  recalled  that  it  is  the  air  above  the  level  ol 
a  given  place  that  produces  the  pressure.  Descending 
a  mountain  or  into  a  mine  simply  submerges  one  more 
deeply  in  the  atmospheric  ocean  and  thus  puts  above  his 
level  more  air  to  be  supported. 
11 


FIG.  110.  —  A  Portion  of  a  Weather  Map. 


146         A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

146.  Atmospheric  Pressure  Computed.  —  In  order  to 
compute  the  pressure  of  the  atmosphere  in  grams  per 
square  centimeter,  we  have  only  to  find  the  pressure  per 
unit  area  due  to  mercury  when  its  depth  is  equal  to  the 
height  of  the  barometric  column.  Consider  a  tube  whose 
cross-sectional  area  is  1  square  centimeter  and  in  which 
the  height  of  mercury  is  76  centimeters.  The  pressure  at 
the  bottom  of  such  a  tube  is  the  product  of  the  area,  height, 
and  density  of  the  mercury,  as  shown  in  §  106.  We  have, 
therefore,  1  x  76  x  13.6,  or  1033.6  grams.  Hence,  if  the 
barometer  reading  is  76  centimeters,  the  atmospheric  pres- 
sure is  1033.6  grams  per"  square  centimeter. 

EXERCISES 

1.  The  column  of  mercury  in  a  barometer  stands  at  a  height  of 
74.5  cm.     What  is  the  height  in  inches? 

2.  How  high  a  column  of  water  could  be  supported  by  atmos- 
pheric pressure  when  the  barometer  reads  75  cm.? 

3.  When  the  barometer  reads  74  cm.,  what  is  the  atmospheric 
pressure  expressed  in  grams  per  square  centimeter?    in  dynes  per 
square  ceptimeter? 

4.  If  the  pressure  of  the  air  is  15  Ib.  per  square  inch,  calculate 
the  total  force  exerted  upon  a  person  the  area  of  whose  body  surface 
is  16  sq.  ft. 

5.  A  soap  bubble  has  a  diameter  of  4  in.     Calculate  the  force 
exerted  by  the  air  against  its  entire  surface  when  the  barometer  reads 

29  in.     A  cubic  inch  of  mercury  weighs  0.49  Ib. 

6.  What   result   would   be   obtained    by   performing 
Torricelli's  experiment  with  a  tube  twice  as  long  as  the 
one  described  in  §  140  ? 

7.  How  would   the  height  of  mercury  be   changed 
if  a  tube  of  larger  cross-sectional  area  were  used  ? 

8.  Show  why  a  change  in  the  area  of  the  mercurial 
surface  in  the  cistern  of  a  barometer  has  no  effect  on  the 
height  of  the  column. 

9.  What  result  would  Torricelli  have  obtained  in  his 
experiment  if  he  had  used  a  tube  only  70  cm.  long? 

FIG.  111.          10-    Try  to  suck  the  water  from  a  bottle  (see  Fig.  Ill) 


MECHANICS   OF  GASES 


147 


FIG.    112.  —  Pneu- 
matic Inkstand. 


out  of  which  a  glass  tube  passes  through  a  tightly  fitting  rubber 
stopper.     Explain  the  results  observed. 

11.  Inkstands  are  sometimes  made  in  the  form 
shown  in  Fig.  112.     Explain  why  the  ink  in  the 
reservoir  can  remain  at  a  greater  height  than  that 
outside. 

12.  Why  is  it  necessary  to  make  a  small  vent 

hole  in  the  upper  part  of  a  cask  when  it  is 
desired  to  draw  off  the  liquid  in  a  steady 
stream  from  a  faucet  placed  near  the  bottom  ? 

13.  Explain  why  water  will  not  run  in 
a  steady  stream  from  an  inverted  bottle. 

14.  The  Magdeburg  hemispheres  shown 
in  Fig.  113  are  closely  fitting  hollow  vessels 
about  4   in.   in  diameter.     By  placing  the 
hemispheres  together   and   exhausting   the 
air  the  pull  of  two  strong  boys  is  scarcely 
sufficient  to  separate  them.     Explain. 

15.  Assuming   the  atmospheric  pressure 
to  be  15  Ib.  per  square  inch,  calculate  the 
force  required  to  separate  the  hemispheres 
shown  in  Fig.  113  when  the  air  within  them 


FIG.     113.  —  Magdeburg 
Hemispheres. 


is  completely  exhausted. 


3.    EXPANSIBILITY  AND  COMPRESSIBILITY  OF  GASES 

147.  Compressibility  of  Air  and  Other  Gases.  —  Gases, 
unlike  liquids,  are  easily  reduced  in  volume  by  increasing 
the  pressure  under  which  they  exist.  This  is  evident 
from  the  fact  that  the  quantity  of  air  in  the  pneumatic  tire 
of  a  bicycle,  for  example,  may  be  increased  to  double  or 
triple  the  original  mass.  Again,  the  air  in  a  pneumatic 
cushion  is  compressed  into  a  smaller  space  when  one  sits 
upon  it,  but  it  springs  back  to  its  original  volume  when 
the  pressure  is  relieved.  Thus  air  and  other  gases  mani- 
fest the  property  of  expansibility  as  well  as  compressibility. 
The  popgun  and  air  rifle  make  use  of  these  properties  of 
air  :  first  the  air  is  compressed  in  the  cylinder  of  the  gun, 


148 


A  HIGH   SCHOOL  COURSE  IN   PHYSICS 


FIG.  114.  -II- 


then  as  the  pellet  moves,  the  force  of  expansion  drives  the 
missile  with  great  acceleration  from  the  barrel. 

1.   Place  a  partially  inflated  balloon  under  the  receiver  of  an  air 
pump.     As  soon  as  the  pump  is  set  in  action,  the  swelling  of  the 
balloon  will  indicate  the  expansive  tendency  of  the  air 
within  it. 

2.  Arrange  a  bottle  as  shown  in  Fig.   114.     Blow 
forcibly  into  the  tube,  thus  causing  some  bubbles  of 
air  to  pass  into  the  bottle  above  the  liquid.     Quickly 
remove   the   lips   from   the  tube, 

and  water  will  be  driven  out  by 
the  expanding  air  within  the 
bottle. 

3.  Place  two  bottles  under  the 
receiver  of  an  air  pump  as  shown 
in  A  and  B,  Fig.  115.    A  is  tightly 

lustrating   corked  and  about  two  thirds  full 

Expansi-   of  Water.    B  is  uncorked  and  con- 

bilitv   of   a 

GagJ  tains  air  only.     When  the  pump 

is  put  into  operation,  the  air  pres- 
sure on  the  surface  of  the  water  in  B  is  reduced, 
and  the  consequent  expansion  of  the  air  in  A  above  the  liquid  forces 
the  water  through  the  tube  into  B.  If  air  be  now  admitted  into  the 
receiver,  the  water  will  be  driven  back  into  A .  Why  ? 

The  expansibility  of  gases  is  explained  by  assuming  that 
their  molecules  (§  129)  are  in  rapid  motion.  As  a  con- 
sequence of  this  motion,  innumerable  molecules  strike 
against  every  part  of  the  walls  of  the  containing  vessel. 
Although  the  mass  of  a  molecule  is  extremely  small,  its 
speed  is  so  enormous  (equaling  or  exceeding  that  of  a 
cannon  ball)  that  it  strikes  the  side  of  the  vessel  with  an 
appreciable  force.  Thus  a  continuous  storm  of  such  blows 
results  in  the  production  of  a  steady  outward  pressure 
against  the  walls  of  the  vessel  inclosing  the  gas. 

148.  Pressure  and  Elastic  Force  in  Equilibrium. —The 
experiments  just  described  teach  us  why  hollow  bodies, 


MECHANICS  OF  GASES 


149 


such  as  balloons,  cardboard  boxes,  etc.,  are  not  crushed  by 
atmospheric  pressure.  The  crushing  force  exerted  against 
the  external  surface  of  a  balloon  only  10  centimeters  in 
diameter  is  more  than  300  kilograms.  This  force,  how- 
ever, is  counteracted  by  the  expansive  force  of  the  gas 
within  it.  If  the  external  force  is  decreased,  the  gas  ex- 
pands until  the  forces  are  again  in  equilibrium.  On  the 
other  hand,  if  the  external  force  is  increased,  the  gas  is 
compressed  until  the  inward  and  outward  forces  balance 
each  other.  Thus  the  human  body  is  able  to  withstand 
the  enormous  force  exerted  by  the  atmosphere  upon  it. 
All  the  cavities  that  might  otherwise  collapse  are  pre- 
vented from  doing  so  by  the  expansive  force  of  the  gases 
which  they  contain. 

149.  Law  of  Expansion  and  Compression. — The  rela- 
tion which  exists  between  the  pressure  to  which  a  gas  is 
subjected  and  its  volume  was 
discovered  by  Robert-  Boyle 
(1627-1691)  of  England,  in 
1662,  and  by  Mariotte,  of 
France,  fourteen  years  later. 
In  France  this  relation  is  usu- 
ally called  "  Mariotte's  Law," 
but  we  speak  of  it  as  Boyle's 
Law.  An  experiment  to  illus- 
trate Boyle's  method  of  dis- 
covery may  be  performed  as 
follows : 


Take  a  bent  glass  tubo  (1),  Fig. 
116,  of  which  the  shorter  arm  is 
hermetically  sealed.  The  long  arm, 
left  open  at  the  top,  should  be  about 
90  centimeters  long.  The  length  FlG 
of  the  short  arm  should  be  at  least 


116.  —  Illustrating    Boyle's 
Law. 


150         A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

10  centimeters.  Pour  a  small  quantity  of  mercury  into  the  tube  so 
that  the  liquid  stands  at  the  same  level  in  both  sides.  We  thus 
have  a  quantity  of  air  confined  in  the  space  A  C  under  one  atmosphere 
of  pressure  owing  to  the  transmission  of  pressure  by  the  mercury. 
Measure  the  length  of  the  space  AC,  and  then  pour  mercury  into 
the  tube  (see  (2),  Fig.  116),  until  the  surface  of  the  liquid  in 
the  long  arm  is  as  far  above  that  in  the  short  arm  as  the  height  of 
mercury  in  the  barometer.  If  the  new  volume  of  air  in  the  short 
arm  be  measured,  it  will  be  found  to  be  just  one  half  the  original 
volume.  The  new  pressure  is  due  both  to  the  atmospheric  pressure 
on  the  mercury  at  B  and  the  column  of  mercury  in  the  tube.  It  is, 
therefore,  two  atmospheres. 

This  experiment  teaches  us  that  by  increasing  the  pres- 
sure upon  a  confined  gas  from  one  atmosphere  to  two 
atmospheres,  the  volume  is  caused  to  be  only  one  half  as 
great.  This  is  only  a  special  instance,  however,  of  a  more 
general  law.  If  p  is  the  pressure  of  a  gas  whose  volume 
is  v,  then 

under  a  pressure  2  p  the  volume  of  the  gas  will  be  |  v  ; 
under  a  pressure  &  p  the  volume  of  the  gas  will  be  J  v  ; 
under  a  pressure  ^p  the  volume  of  the  gas  will  be  2  v  ; 
under  a  pressure  ^p  the  volume  of  the  gas  will  be  3  v,  etc. 

Now,  since 

pv  =  2p  x  I ,v  =  3p  x  \v  =  \p  x  2  v  =  ±p  x  3  v, 
we  are  able  to  state  the  relation  as  follows : 

The  product  of  the  pressure  and  volume  of  a  given  mass  of 
gas  at  a  constant  temperature  is  constant. 

Where  P  is  the  pressure  of  the  gas  when  its  volume  is 
FJ  and  p  the  pressure  when  the  volume  is  v,  the  law  may 
be  expressed  algelxtfifelly  in  the  forms : 

PV  =  pv,  or  P  :  p  : :  v  :  V.  (l) 

EXAMPLE.  —  The  observed  volume  of  a  gas  is  30  cm.3  when  the 
barometer  reads  74.5  cm.  What  volume  will  it  occupy  under  a  baro- 
metric pressure  of  76  cm.? 


MECHANICS   OF  GASES 


151 


SOLUTION.  —  According  to  Boyle's  Law  the  product  of  the  pres- 
sure and  volume  under  the  first  condition  will  be  equal  to  their  prod- 
uct under  the  second  condition.  Hence,  if  x  represents  the  volume 
required,  we  have 

74.5  x  30  =  76  x  x. 

Solving  this  equation  for  x,  we  obtain 

x  =  29.4  cm.3. 

EXERCISES 

1.  If  the  volume  of  a  certain  gas  is  200  cm.3  when  its  pressure  is 
1000  g.  per  square  centimeter,  what  volume  will  it  occupy  when  its 
pressure  has  been  increased  to  1200  g.  per  square  centimeter? 

2.  The  volume  of  an  air  bubble  at  a  depth  of  1  m.  of  mercury  is 
1  cm.3     What  will  be  its  volume  when  it  reaches  the  surface  if  the 
barometer  reading  is  75  cm.  ? 

SUGGESTION.  —  The  first  pressure  is  equal  to  the  sum  of  the  atmos- 
pheric pressure  and  that  due  to  the  mercury  in  which  it  is  immersed, 

or  175  cm., 

« 

3.  A  gas  is  often  confined  in  a  tube,  as  shown 
in  Ay  Fig.  117,  whose  open  end  is  beneath  the  sur- 
face of  some  liquid.     How  much  is  the  pressure  of 
the  gas  confined  in  such  a  tube  in  a  vessel  of  mer- 
cury when  the  surface  in  the  tube  is  25  cm.  below 
the  level  of  the  liquid  outside,  the  barometer  read- 
ing 75  crn.? 

4.  If  the  volume  of  the  gas  under  the  conditions 
given  in  Exer.  3  is  15  cm.,  what  will  be  its  volume 
if  the  tube  is  elevated  until  the  surfaces  are  at  the 
same  level  ? 

5.  In  B,  Fig.  117,  the  surface  of  the  mercury  in 
the  tube  is  25  cm.  above  that  of  the  mercury  on 
the  outside.    If  the  atmospheric  pressure  is  75  cm., 
what  is  the  pressure  of  the  confined  gas  ? 

6.  Under  the  conditions  given  in  Exer.  5  the  volume  of  the  gas 
confined  in  the  tube  is  50  cm.3.     What  volume  will  the  gas  occupy 
when  the  surfaces  are  brought  to  the  same  level? 

7.  The  volume  of  an  air  bubble  136  cm.  under  vater  is  0.5  cm.3. 
Barometer  reading  =  75.4  cm.     Calculate  (1)  the  pressure  to  which 
the  bubble  is  subjected,  and  (2)  the  volume  it  will  have  as  it  emerges 
from  the  water. 


FIG.  117. 


152         A  HIGH   SCHOOL  COURSE   IN    PHYSICS 

SUGGESTION.  —  Reduce  the  depth  of  water  to  its  equivalent  in 
terms  of  mercjiry,  and  compute  the  first  pressure  as  in  Exer.  2. 

8.  To  what  depth  would  the  inverted  tumbler  shown  in  Fig.  1 
have  to  be  taken  in  order  to  become  half  filled  with  water  if  the 
barometer  reading  is  75  cm.  ? 

9.  A  gas  tank  whose  capacity  is  3.5  cu.  ft.  is  filled  with  illumi- 
nating gas  until  the  pressure  is  225  Ib.  per  square  inch.     How  many 
cubic  feet  of  gas  at  atmospheric  pressure  will  be  required  in  the  filling 
of  the  tank?     (Assume  one  atmosphere  to  be  15  Ib.  per  square  inch.) 

10.  Why  is  it  safer  to  test  an  engine  boiler  by  pumping  in  water 
rather  than  air?l^%^ 

150.  Changes  in  Density.  —  Since  a  change  in  the  volume 
of  a  given  mass  of  gas  occurs  whenever  the  pressure  is 
changed,  it  follows  that  there  will  also  be  a  change  in 
its  density.  Forcing  the  gas  into  one  half  its  original 
volumeApBubles  the  amount  in  each  cubic  centimeter  ; 
makhiJMthe  volume  one  third  multiplies  the  mass  in  each 
cubic  centimeter  by  three  ;  and  so  on.  Therefore,  doub- 
ling the  pressure  of  a  gas,  thus  reducing  the  volume  one 
half,  doubles  the  density  ;  tripling  the  pressure  multiplies 
the  density  by  three  ;  and  so  on.  Hence,  we  may  state 
these  relations  as  follows  : 

The  density  of  a  gas  at  constant  temperature  is  directly 
proportional  to  the  pressure. 


Expressed  algebraically, 

D  :  d  :  :  P  :  p,  (2) 

where  D  is  the  density  of  the  gas  when  the  pressure  is  P, 
and  d  the  density  when  the  pressure  is  p. 

EXERCISES 

1.  Hydrogen,  Vhose  density  is  0.09  g.  per  liter  under  one  atmos- 
phere of  pressure,  is  condensed  in  a  steel  cylinder  until  the  pressure 
is  15  atmospheres.  Calculate  the  density  of  the  gas  in  the  cylinder. 


MECHANICS   OF  GASES  153 

2.  Illuminating  gas  is  condensed  in  a  reservoir  until  its  density 
has  increased  from  0.75  g.  per  liter  to  4.5  g.  per  liter.     Calculate  the 
pressure  in  the  reservoir.     Express  the  result  in  atmospheres. 

3.  If  4  liters  of  air  at  ordinary  atmospheric  pressure  are  admitted 
into  a  vacuum  of  10  liters  capacity,  what  will  be  the  pressure  and 
density  of  the  air?    (Under  one   atmosphere   the   density  of  air  is 
1.29  g.  per  liter.) 

4.  What  is  the  weight  of  the  quantity  of  illuminating  gas  con- 
densed in  a  cylindrical  tank  of  3  cu.  ft.  capacity  until  the  pressure  is 
225  Ib.  per  square  inch  ?     (The  density  of  the  gas  under  one  atmos- 
phere of  pressure  is  0.75  g.  per  liter.) 

5.  If  the  gas  shown  in  the  tubes  in  Fig.  117,  A  and  B,  is  air,  what 
is  the  density  of  it  under  the  conditions  given  in  Exercises  3  and  5 
on  page  151  ? 

4.    ATMOSPHERIC   DENSITY   AND    BUOYANCY 

151.    Atmospheric  Density  Changes  with  Altitude.  —  The 

air,  unlike  the  water  of  the  ocean,  which  is  practically 
incompressible,  diminishes  in  density  as  one^  ascends  a 
mountain  or  rises  in  a  balloon.  As  the  pressure  becomes 
less,  the  density  of  the  air  decreases  proportionally.  Thus 
at  the  summit  of  Mont  Blanc  in  Switzerland,  an  altitude 
of  three  miles,  the  barometer  indicates  only  one  half  as 
much  atmospheric  pressure  as  at  the  sea-level.  Hence  the 
density  of  the  air  at  "this  altitude  is  only  one  half  as 
great. 

Aeronauts 'have  succeeded  in  ascending  to  an  altitude 
of  about  7  miles,  where  the  pressure  is  only  18  centimeters, 
or  about  a  quarter  of  sea-level  pressure.  Greater  altitudes 
have  been  explored  by  the  aid  of  balloons  equipped  with 
self-registering  instruments  until  a  height  of  about  14 
miles  has  been  attained.  Figure  118  shows  the  changes  in 
the  pressure  and  density  of  the  atmosphere  at  various  alti- 
tudes. The  numbers  at  the  left  indicate  altitudes  in  miles 
above  sea-level,  those  at  the  extreme  right  the  densities  of 
the  air  compared  with  the  density  at  sea-level,  while  the 


154 


A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


next  column  gives  the  barometric  readings  in  inches.  From 
the  figure  it  may  be  seen  that  at  an  elevation  of  15  miles, 
for  example,  the  density  of  the  air  is  only  one  thirtieth  of 


FIG.  118.  —  Showing  the  Decrease  of  Atmospheric  Pressure  with  Altitude. 

its  sea-level  density,  while  the  barometer  would  read  about 
one  inch  of  mercury. 

152.  Buoyancy  of  Air.  — »•  On  account  of  the  fact  that 
the  pressure  of  the  air  decreases  as  the  altitude  increases, 
its  pressure  downward  upon  the  top  surface  of  a  box,  for 
example,  is  less  than  its  upward  pressure  against  the  bot- 
tom. It  follows,  therefore,  as  for  bodies  immersed  in 
water  (§  121),  that  an  object  is  buoyed  up  by  a  force  equal 
to  the  weight  of  the  air  that  it  displaces.  In  general, 
bodies  are  so  heavy  in  comparison  with  the  amount  of  air 
displaced  that  the  consequent  loss  of  weight  is  not  taken 
into  account.  A  man  of  average  size,  for  instance,  is 
buoyed  up  by  a  force  equal  to  about  4  ounces.  If,  how 


MECHANICS   OF  GASES  155;. 

ever,  the  air  displaced  by  a  body  should  weigh  more  than 
the  body  itself,  it  would  be  lifted  by  the  air  just  as  a  piece 
of  wood  is  lifted  when  immersed  in  water.  This  is  the 
case  of  a  balloon,  in  wln'ch  the  weight  of  the  material  com- 
posing the  bag,  together  with  the  gas  used  to  fill  it,  ropes, 
basket,  ballast,  etc.,  is  less  than  that  of  the  air  which  it 
displaces.  In  order  that  this  may  be  so,  it  is  necessary 
to  inflate  the  balloon  with  a  gas  lighter  than  air,  which 
weighs  1.29  kilograms  per  cubic  meter.  Hydrogen  gas  is 
sometimes  used  whose  density  is  0.09  kilogram  per  cubic 
meter,  but  more  frequently  the  material  is  common  illumi- 
nating 'gas  weighing  about  0.75  kilogram  per  cubic  meter. 
The  balloon  United  States  which  won  the  first  international 
race  at  Paris  in  1906  was  filled  with  over  2000  cubic  meters 
of  illuminating  gas,  thus  creating  a  lifting  force  of  more 
than  a  ton. 

EXERCISES 

1.  A  balloon  whose  capacity  is  1000  m.3  is  filled  with  hydrogen. 
If  the  weight  of  the  bag,  basket,  and  ropes  is  235  kg.,  what  additional 
weight  can  the  balloon  lift  ? 

SUGGESTION.  —  Find  the  difference  between  the  weight  of  the  air 
displaced  by  the  balloon  and  the  entire  weight  of  the  balloon  and  its 
equipment. 

2.  What  will  be  the  lifting  capacity  of  the  balloon  in  Exer.  1  when 
filled  with  illuminating  gas? 

3.  When  will  a  balloon  cease  to  rise  ?     When  will  it  begin  to  fall  ? 

4.  A  kilogram  weight  of  brass  (density  8.3  g.  per  crn.3)  will  weigh 
how  much  in  a  vacuum  ? 

SUGGESTION.  —  Add  to  the  weight  of  the  body  the  weight  of  the  air 
that  it  displaces. 

5.   APPLICATIONS    OF   AIR   PRESSURE 

153.  The  Air  Pump.  —  The  air  pump  is  used  to  remove 
the  air  or  other  gases  from  a  closed  vessel  called  a  receiver. 
It  was  invented  about  1650  by  Otto  von  Guericke  (1602- 


156 


A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


1686),  burgomaster  of  Magdeburg,  Germany.  A  simple 
form  of  the  air  pump  is  shown  in  Fig.  119.  O  is  a  cylin- 
der within  which  slides  the  tightly  fitting  piston  P.  R  is 
the  receiver  from  which  the  air  is  to  be  exhausted.  The 
receiver  is  connected  with  the  cylinder  by  the  tube  T. 
8  and  t  are  valves  opening  upward.  The  operation  of 
the  pump  is  as  follows  : 

When  the  piston  is  pushed  down,  the  valve  s  permits 
the  air  in  the  cylinder  to  escape,  but  closes  to  prevent  its 

return  when  the  piston 
is  lifted.  Raising  the 
piston  tends  to  produce 
a  vacuum  in  the  cylin- 
der ;  but  the  air  in  the 
receiver  and  connecting 
tube  expands,  lifts  the 
valve  £,  and  fills  the 
cylinder.  Thus  each 
down-and-up  stroke  of 
the  piston  results  in  the 
removal  of  a  portion  of 
the  air  in  the  receiver.  After  several  strokes  of  the 
piston,  the  air  in  the  receiver  becomes  rarefied  to  such  an 
extent  that  its  expansive  force  is  no  longer  sufficient  to 
lift  valve  £,  and  no  further  exhaustion  can  be  produced. 

Some  air  pumps  are  so  constructed  that  the  valves  are 
opened  and  closed  automatically ~at  the  proper  moment  so 
that  a  greater  degree  of  rarefaction  can  be  reached. 

154.  The  Condensing  Pump.  —  The  condensing  pump  is 
used  to  compress  illuminating  gases  in  cylinders  for  use  in 
lighting  vehicles,  stereopticons,  etc.,  and  further  for  inflat- 
ing pneumatic  tires,  operating  drills  in  mines,  air-brakes 
on  railway  cars,  and  for  many  other  purposes.  The  most 
common  condensing  pump  is  that  used  for  inflating  bicycle 


FIG.  119.  —  Air  Pump. 


MECHANICS   OF  GASES 


157 


tires.  Figure  120  shows  the  construction  of  such  a  pump. 
When  the  piston  P  is  forced  down,  the  air  in  the  cylindei 
of  the  pump  is  driven  into 
an  air-tight  rubber  bag 
within  the  tire  T.  The 
small  valve  8  opens  to  ad- 
mit the  air  into  the  tire, 
but  closes  to  prevent  its 
return.  On  lifting  the 
piston  a  partial  vacuum  is 
produced  in  the  cylinder, 

and    the   air    from    outside    FIG.  120.  -  Inllatlng  a  Tire  r  by  Means 

of  a  Condensing  Pump  P. 

finds  its  way  into  the  cyl- 
inder past  the  soft  cup-shaped  piece  of  leather  attached  to 
the  piston.  During  the  downward  stroke  of  the  piston, 
however,  this  leather  is  pressed  firmly  against  the  sides 
of  the  cylinder  and  thus  prevents 
the"  escape  of  the  air.  Repeated 
strokes  of  the  piston  add  to  the 
mass  of  air  already  in  the  tire,  and 
the  process  of  pumping  may  be  con- 
tinued until  the  tire  is  sufficiently 
inflated. 

155.  The  Common  Lift  Pump.  — 
The  simplest  pump  for  raising  water 
from  wells  is  the  common  lift  pump. 
This  consists  of  a  barrel,  or  cylinder, 
(7,  Fig.  121,  connected  with  a  well 
or  other  source  of  water  by  a  pipe 
B.  The  entrance  of  this  pipe  into 
the  cylinder  is  covered  by  a  valve 
«,  opening  upward.  In  the  cylinder 
is  a  closely  fitting  piston  P  which 

FIG.    121.  —  Common    Lift  ,  .      , 

Pump.  can  be  raised  or  lowered  by  means 


158 


A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


of  a  rod  which  is  usually  connected  with  a  lever  for  con- 
venience in  operating.  The  piston  contains  a  valve  t,  also 
opening  upward.  The  action  of  the  pump  is  as  follows : 
When  the  piston  is  raised,  the  pressure  of  the  air  in  the 
tube  and  lower  part  of  the  cylinder  is  diminished.  The 
atmospheric  pressure  on  the  surface  of  the  water  in  the 
well  then  forces  water  into  the  pipe.  As  the  piston  is 
lowered,  valve  s  closes,  and  t  allows  the  air  in  the  cylinder 
to  escape,  but  closes  again  when  the  piston  begins  to  ascend. 
The  second  stroke  again  reduces  the  pressure  below  the 
piston,  and  water  is  forced  still  higher  in  the  pipe.  At 
last  the  water  reaches  the  piston,  passes  through  during 
the  downward  stroke,  and  is  lifted  toward  the  spout  of  the 
pump  when  the  piston  moves  upward. 

It  will  be  seen  that  the  action  of  the  common  lift  pump 
is  dependent  on  the  pressure  of  the  atmosphere  to  elevate 
the  water  to  a  point  just  high  enough  to  come  within  reach 
of  the  piston  ;  the  piston  must  not  be  farther  above  the  water 

in  the  well  than  the  height  of 
the  water  column  that  the  air 
can  support.  When  water 
is  to  be  lifted  from  a  well 
more  than  33  or  34  feet  in 
depth,  the  cylinder  is  placed 
low  enough  to  enable  the 
piston  to  move  within  that 
distance  of  the  surface  of 
the  water. 

156.  The  Force  Pump.  — 
The  force  pump  is  used  to 
deliver  water  under  consid- 
erable pressure  either  for 
spraying  purposes  or  in  or- 
FIG.  122.— A  Force  Pump.  der  to  elevate  it  to  a  reser- 


MECHANICS   OF  GASES  159 

voir  placed  some  distance  above  the  level  of  the  pump.  It 
is  made  in  many  different  forms.  Usually  the  force  pump 
has  no  opening  or  valve  in  the  piston.  The  water  escapes 
from  the  cylinder  through  a  side  opening  A^  Fig.  122,  past 
the  valve  £,  thence  upward  to  the  spout  or  reservoir.  In 
other  respects  the  description  of  the  action  of  the  lift  pump 
applies  equally  well  to  the  force  pump.  In  order  to  obtain 
a  steady  stream  of  water,  a  force  pump  is  often  provided 
with  an  air  chamber  D.  The  entrance  of  the  water  through 
t  serves  to  compress  the  air  in  the  chamber,  which  by  its 
expansive  force  maintains  the  current  in  pipe  E  while  the 
piston  is  moving  upward. 

157.  The  Siphon.  —  The  siphon  is  a  bent  tube  with  un- 
equal arms.  It  is  used  for  removing  liquids  from  tanks  or 
reservoirs  that  have  no  outlet,  or  for  P.a6 
drawing  off  the  liquid  from  a  vessel 
without  disturbing  a  sediment  lying 
upon  the  bottom. 

Let  a  glass  tube  bent  in  the  form  shown  in 
Fig.  123  be  filled  with  water,  and  the  ends 
closed  while  the  tube  is  inverted  and  placed 
in  the  position  shown.  On  opening  the  ends 
water  will  flow  through  the  tube  from  the 
vessel  in  which  the  liquid  has  the  higher 
level. 

The  action  of  the  siphon  is  explained 
as  follows  :  the  upward  pressure  at  a 
due  to  the  atmosphere  is  only  partly 
counterbalanced  by  the  pressure  of  the  FlG'  123-~The  SiPhon- 
liquid  column  ah.  If  the  atmospheric  pressure  is  p,  the 
resultant  force  acting  toward  the  right  is  p  —  db.  Again, 
the  upward  pressure  of  the  atmosphere  at  d  is  opposed 
by  the  pressure  due  to  the  liquid  column  cd.  Hence 
the  resultant  pressure  acting  toward  the  left  is  p  —  cd. 


160         A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

Since  the  pressure  due  to  the  liquid  in  the  long  arm 
exceeds  that  of  the  liquid  in  the  short  arm,  the  excess 
of  force  in  the  tube  tends  to  move  the  water  toward 
the  lower  vessel.  The  resultant  of  all  the  forces  is 
equal  to  the  difference  of  pressure  due  to  the  liquids  in 
the  two  arms,  i.e.  to  a  column  of  liquid  equal  to  cd  —  ab. 
The  liquid  will  therefore  continue  to  flow  until  this  dif- 
ference is  0 ;  or,  in  other  words,  until  the  liquid  reaches 
the  same  level  on  the  two  sides.  A  second  condition 
necessary  to  the  action  of  a  siphon  is  that  for  water  the 
height  of  the  lend  f  must  not  be  greater  than  33  or  34  feet 
above  a,  since  its  elevation  to  the  point  /  is  due  to  the 
atmospheric  pressure. 

158.  Work  done  by  Compressed  Gases.  —  Since  a  gas 
exerts  a  force  against  the  sides  of  the  vessel  containing  it, 
it  is  obvious  that  work  will  be  done  by  the  gas  if  the  side 
of  the  vessel  is  made  movable  and  the  gas  allowed  to  ex- 
pand (§  55).  Imagine  air  to  be 
compressed  in  the  cylinder  O  shown 
in  Fig.  124,  in  which  P  represents 
FIG.  124.  — An  Expanding  a  movable  piston.  If  the  external 
resistance  offered  to  the  piston  is 
not  as  great  as  the  force  exerted 
by  the  air  inside,  work  will  be  done  by  the  expanding 
gas,  and  the  piston  will  move.  If  the  pressure  remains 
constant  during  the  process,  the,  work  done  will  be 
measured  by  the  product  of  the  force  exerted  by  the  air 
against  the  piston  and  the  displacement  produced. 

EXAMPLE.  —  A  tube  leads  from  a  cylinder  to  a  reservoir  where  air  is 
stored  under  a  pressure  of  5000  g.  per  cm.2;  the  area  of  the  piston  is  50 
cm.2.  Find  the  work  done  by  the  gas  when  the  piston  moves  20  cm. 

•  SOLUTION.  —  The  total  force  exerted  by  the  gas  against  the  piston 
is  50  x  5000,  or  250,000  g.  Hence  the  work  done  is  250,000  x  20,  or 
5,000,000  gram-centimeters. 


MECHANICS   OF  GASES 


161 


Since  energy  can  be  transferred  from  place  to  place  in 
a  compressed  gas,  air  under  great  pressure  is  often  con- 
ducted through  pipes  over  long  distances  from  a  condens- 
ing apparatus  (§  154)  to  a  point  where  the  energy  is  to  be 
utilized.  In  this  way  a  locomotive  engineer  is  able  to  con- 
trol a  train  by  means  of  compressed  air  conducted  from 
the  engine,  where  it  is  compressed,  to  suitable  apparatus 
in  connection  with  the  brakes  under  each  car  (§  159). 
In  mining  operations  and  stone  quarrying,  pneumatic  ma- 
chines utilize  compressed  air  for  drilling  the  holes  in  rocks 
where  explosives  used  in  blasting  are  to  be  placed.  Many 
other  applications  of  the  power  of  compressed  gases  have 
found  a  place  in  modern  engineering. 

159.    The  Air  Brake.  —  Compressed  air  is  widely  used  by  the 
railroads  of  many  countries  in  the  operation  of  the  Westing-house  air 
brake,  Fig.  125,  which  works  as  follows :  —  The  locomotive  is  provided 
with  a  condensing  pump 
which  keeps  the  air  in  a 
large  reservoir  at  a  pres- 
sure of  about  80  pounds 
per  square  inch.      From 
this  reservoir    the    com- 
pressed  air  is  conducted 
through  the  train  pipe  P 
to  an  auxiliary  reservoir  To  the 
R  placed  under  each  car.  Brake 
As  long  as  the  pressure  is 
maintained   in   P,  air   is       FlG  125  _The  Westinghouse  Air  Brake. 
allowed  to  enter  R  and  at 

the  same  time  is  prevented  from  entering  the  cylinder  C  by  a  compli- 
cated automatic  valve  F,  and  the  brakes  are  held  "off"  by  the  spring 
S.  If,  however,  the  engineer  by  moving  a  lever  in  the  cab,  allows  the 
pressure  in  the  pipe  P  to  fall,  the  passage  between  R  and  P  is  at 
once  closed  by  the  v^lve  F,  and  the  compressed  air  in  R  is  admitted 
into  C.  The  pressure  of  the  air  forces  the  piston  to  the  left  and 
sets  the  brakes  against  the  wheels.  By  admitting  air  from  the  reser- 
voir on  the  engine  into  the  pipe  P,  the  valve  V  again  establishes  a 
12 


162 


A  HIGH  SCHOOL  COURSE   IN   PHYSICS 


communication  between  P  and  R  and  allows  the  air  in  C  to  escape. 

The  spring  S  forces  the  piston  back  and  releases  the  brakes.     The 

great  advantage  of  this  brake  is 
that  in  case  of  any  accidental 
breaking  of  the  train  pipe  P, 
the  brakes  are  automatically  set. 
They  are  often  arranged  to  be 
operated  from  any  coach  in  case 
of  emergency. 

160.  Subaqueous  Opera- 
tions. —  Important  use  is  made 
of  compressed  air  in  various  en- 
gineering operations  performed 
under  water,  as  laying  the  foun- 
dations of  bridges,  excavating  for 
tunnels,  recovering  the  cargoes 
from  sunken  vessels,  etc.  Figure 
126  shows  the  most  important 
apparatus  for  subaqueous  work, 
the  diving  bell,  and  the  sub- 
marine diver.  A  condensing 

pump  on  board  the  vessel  forces  air  through  a  tube  into  the   bell, 

thus   supplying  the  workman  with  oxygen  and  preventing  the  rise 

of  water  in  the  working  chamber. 

Air  is  supplied  to  the  workman 

in    diving   armor   in    the   same 

manner.     The  foul   air  escapes 

from  the  diving  suit  through  a 

valve  above  the  chest.     In  many 

cases  the  air  is  supplied  from  a 

reservoir  carried  on  the  back  of 

the  diver. 

Deep  excavations  are   fre- 
quently  made   and   foundations 

built  up  from  bed  rock  by  the 

aid    of    the    pneumatic   caisson 

(pronounced  kas'son).     See  Fig. 

127.     The   working  chamber  C  BedKock 

is  gradually  lowered  through  soft    FlG  127  _  Excavating  below  the  Water 

soil  by  removing  the  earth  from         Level  by  the  Aid  of  Compressed  Air. 


FIG.  12G.  —  Apparatus  Used  for  Work 
under  Water. 


MECHANICS   OF  GASES  163 

within  after  loading  the  roof  of  the  chamber  with  heavy  masonry 
above.  When  the  caisson  sinks  below  water  level,  air  under  suitable 
pressure  is  forced  into  C  to  prevent  the  influx  of  water.  The  ex- 
cavated earth  is  removed,  and  the  foundation  material  introduced 
through  the  air  lock  A.  The  bucket  is  lowered  into  A,  after  which 
the  opening  around  the  wire  cable  is  closed  air-tight.  Air  is  then 
allowed  to  flow  into  A  until  the  pressure  there  is  equal  to  that  in  (7. 
The  semicircular  doors  separating  A  and  C  are  now  thrown  open,  and 
the  bucket  lowered  into  the  caisson.  As  the  caisson  is  forced  more 
and  more  below  water  level,  the  pressure  of  the  air  is  correspondingly 
increased  in  order  to  prevent  the  water  and  rnud  from  crowding  in. 

EXERCISES 

1.  If  the  pressure  against  the  8-inch  piston  of  an  air  brake  is  75  Ib. 
per  square  inch,  how  much  force  drives  the  piston  forward? 

2.  When  a  train  is  broken  in  two,  the  cars  are  brought  quickly  to 
rest.     Explain. 

3.  What  are  the  advantages  gained  by  the  use  of  air  brakes? 

4.  A  diver  sinks  68  ft.  below  the  surface  of  water.     Under  how 
many  atmospheres   is  he  working?     Explain   why  his  body  is  not 
crushed  by  this  force. 

5.  A  caisson  is  sunk  until  the  bottom  is  51  ft.  below  water  level. 
Under  what  pressure  must  the  laborers  work? 

6.  Explain  how  a  bucket  of  earth  is  removed  from  a  caisson  through 
the  air  lock. 

4  SUMMARY 

1.  The  laws  of  pressure  relative  to  liquids  are  equally 
applicable  to  gases,  except  that  the  pressure  is  not  propor- 
tional to  the  depth  (§§  135  and  136). 

2.  The  density  of  the  air  under  standard  conditions  of 
pressure  and  temperature  (i.e.  76  cm.  and  0°  C.)  is  1.293 
g.  per  liter  or  about  1.25  oz.  per  cubic  foot  (§  137). 

3.  The  pressure  of  the  atmosphere  is  measured  by  the 
barometer.     At  sea  level  the  average  height  of  the  mer- 
curial column  supported  by  the  air  is  76  cm.  or  very  nearly 
30  in.     Expressed  in  units  of  force,  the  average  sea-level 
pressure  is  1033.6  g.  per  square  centimeter  or  14.7  Ib.  per 


164         A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

square  inch.     The  atmospheric  pressure  decreases  with 
the  altitude,  but  not  proportionally  (§§  139  to  146). 

4.  Gases  adapt  themselves  to  the  form  and  capacity  of 
the  vessels  containing  them.     The  expansive  force  of  a  gas 
confined  in  a  receptacle  tends  to  prevent  the  collapse  of  the 
vessel  under  atmospheric  pressure  (§  147). 

5.  The  product  of  the  pressure  and  volume  of  a  given 
mass  of  a  gas  at  a  constant  temperature  is  constant.     This 
is  known  as  Boyle's  Law,  or  Mariotte's  Law  (§  149). 

6.  The  density  of  a  gas  at  a  constant  temperature  is 
directly  proportional  to  the  pressure  to  which  it  is  subjected 
(§150). 

7.  A  body  in  air  is  buoyed  up  by  a  force  equal  to  the 
weight  of  the  air  that  it  displaces  (§  152). 

8.  The  air  pump  is  used  in  the  rarefaction  of  gases. 
The  condensing   pump  is  used  in  compressing   air   and 
other  gases  (§§  153  and  154). 

9.  The  action  of  "  suction  "  pumps  is  dependent  upon 
the  pressure  of  the  atmosphere  on  the  water  in  the  well  or 
cistern  (§§  155  and  156).  „ 

10.  The  siphon   is  a  bent  tube  .having  unequal  arms 
used  for  conveying  a  liquid  over  the  side  of  a  reservoir  to 
a  lower  level  than  that  in  the  reservoir.      Its   action  is 
dependent  on  atmospheric  pressure  (§  157). 

11.  Compressed  air  possesses  energy.     This  energy  is 
used  in  many  important  mechanical  devices,  as  the  air 
brake,  rock  drills,  etc.  (§  158). 


CHAPTER  IX 


SOUND :   ITS  NATURE  AND  PROPAGATION 
1.  ORIGIN  AND  TRANSMISSION  OF  SOUND 

161.  Cause  of  Sound.  —  Whenever  the  sensation  of  sound 
is  traced  to  its  external  cause,  we  find  that  its  source  is 
always  something  which  is  in  a  state 
of  vibration.  Sometimes  the  vibra- 
tion of  the  body  emitting  the  sound 
is  sufficiently  great  to  be  visible,  i.e. 
to  give  a  certain  blurred  indistinct- 
ness to  the  outline  of  the  body.  This 
is  easily  perceived  when  a  stretched 
wire  is  plucked  or  a  tuning  fork  is 
sounded. 


FIG.  128.— Demonstrating 
the  Motion  in  a  Sound- 
ing Tuning  Fork. 


Let  a  small  pith  ball  suspended  on  a  thread 

be  allowed  to  touch  a  tuning  fork  that  is 

emitting  a  sound.     (See  Fig.  128.)     It  will  be  thrown  violently  away. 

Touch  one  of  the  prongs  lightly  to  the  water  in  a  tumbler.  A  ripple 

is  produced,  or  perhaps  a 
spray  is  thrown  from  the 
prong.  Again,  attach  a 
fine  wire  or  bristle  to  the 
prong  of  a  tuning  fork  with 
a  small  quantity  of  sealing 
wax.  Sound  the  fork,  and 


FIG.  129.— The  Motion  in  the  Prong  of  a  Fork 
is  Vibratory. 


draw  the   bristle  across  a 


piece  of  smoked  glass.     A 
wavy  line,  Fig.  129,  results, 
showing  the  existence  of  a  back-and-forth  motion  in  the  fork. 


162.    The  Nature  of  a  Vibration. — The  vibration  of  a 
pendulum  has  been   studied   in    §  80.    *  While   sounding 

165 


166          A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

bodies  vibrate  in  a  manner  similar  to  that  of  the  pendu- 
lum, they  vibrate  from  a  different  cause.  When  a  pendu- 
lum is  drawn  aside  and  then  set  free,  a  component  of  the 
force  of  gravity  moves  the  pendulum  bob  back  toward  its 
original  position  at  the  center  of  its  arc.  When,  how- 
ever, we  pluck  the  string  of  a  guitar,  for  example,  or  set 
the  prongs  of  a  tuning  fork  in  vibration,  the  motion  is  due 
to  the  elasticity  of  the  material  of  which  the  body  is  com- 
posed. The  string,  on  being  drawn  aside,  is  stretched 
slightly.  Now,  on  being  released,  its  tendency  to  resume 
its  original  length  causes  it  to  straighten.  When  we  set 
a  tuning  fork  in  vibration,  the  prongs  are  bent.  Since 
they  are  made  of  elastic  steel,  they  immediately  tend  to 
resume  their  original  shape.  Hence  the  prongs  move 
back  to  their  initial  positions.  Again,  the  string  of  the 
guitar,  like  the  pendulum,  is  in  the  state  of  motion  when 
it  reaches  its  original  position.  Hence  it  must  continue 
to  move  until  some  resistance  checks  it.  Therefore  it 
swings  beyond  this  position  to  a  point  where  its  velocity 
is  zero,  whence  it  returns  in  the 
same  manner  as  before.  The  phe- 
nomenon is  repeated  by  the  string 
until,  like  the  pendulum  again,  its 
energy  is  expended  in  overcoming 
the  resistance  of  the  air  and  the 
friction  of  its  own  molecules. 

Clamp  a  thin  strip  of  wood,  as  a  yard- 
stick, by  one  end,  Fig.  130,  and  set  it  in 
vibration.  As  it  is  being  drawn  aside,  its 
tendency  to  move  back  toward  its  original 

FIG.    130. -Thin    Strip   of    Position  is  veiT  apparent.     Let   the  stick 
Wood  in  Vibration.  move  very  slowly  back  to  the  original  posi- 

tion, and  it  will  stop  there ;  but  if  it  is  set 

entirely  free,  it  will  continue  to  move  until  it  is  some  distance  beyond 
the  center.     Why  ?    Attach  a  weight  of  100  or  200  grains  near  the 


SOUND:  ITS  NATURE   AND   PROPAGATION       167 

free  end  of  the  stick,  and  it  will  be  found  to  vibrate  much  more  slowly 
than  before.  The  force  due  to  the  elasticity  of  the  wood  cannot  bring 
the  increased  mass  so  quickly  to  the  center  nor  stop  it  so  quickly 
when  the  center  is,  passed  on  account  of  the  increased  inertia.  Make 
a  comparison  between  the  vibrating  yardstick  and  the  sounding  tuning 
fork.  The  yardstick  does  not  vibrate  with  sufficient  rapidity  to  pro- 
duce sound. 

163.  Transmission  of  Sounds  to  the  Ear.  —  Sounds  reach 
the    ear   through   the    air    as    the   transmitting   medium. 
Many  other  substances  may,  however,  be  the  means  of 
propagation,  as  the  following  experiments  will  show : 

1.  Let  the  ear  be  held  against  one  end  of  a  long  bar  of  wood,  and 
let  the  shank  of  a  small  vibrating  tuning  fork  be  brought  against  the 
other  end.     A  loud  sound  will  be  heard.     The  scratch  of  a  pin  at  one 
end  can  easily  be  heard  at  the  other. 

2.  Place  the  shank  of  a  tuning  fork  in  a  hole  bored  in  a  large  cork. 
Set  a  tumbler  full  of  water  upon  a  resonance  box,  -and  bring  the  cork 
in  contact  with  the  surface  of  the  water.     If  the  fork  is  in  vibration, 
the  water  will  transmit  the  motion  to  the  box,  as  shown  by  the  in- 
creased intensity  of  the  tone.    Again,  as  most  boys  have  found  experi- 
mentally, if  the  ear  be  held  beneath  the  surface  when  two  stones  are 
struck  together  under  water,  a  loud  sound  results  even  at  some  dis- 
tance from  the  stones. 

164.  Medium  Necessary  for  Propagation  of  Sound.  —  That 
a  sounding  body  cannot  be  heard  without  the  presence  of 
some  transmitting  medium  may  be  shown 

by  the  aid  of  the  air  pump. 

Let  an  electric  bell  be  placed  upon  a  thick  pad  of 
felt  or  cotton,  suspended  by  flexible  wire  springs 
under  the  receiver  of  an  air  pump,  as  shown  in  Fig. 
131.  Make  the  connection  with  a  battery,  so  that 
the  bell  can  be  rung  from  the  outside.  Set  the  bell 
ringing,  and  begin  to  exhaust  the  air  from  the  re- 
ceiver. The  sounds  coming  from  the  bell  become  FlGt  131-~~ Bel1 
,  IT  T  •  ,.i  Ti  Ringing  in  a 

less  and  less  distinct  until  the  greatest  possible  ex-          Partial    Vac- 

haustion  has  been  produced.     If  now  the  air  is  slowly         uuin. 


168          A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

admitted  into  the  receiver,  the  loudness  of  the  sound  increases  until 
its  full  intensity  is  reached. 

Although  the  sound  of  the  bell  used  in  this  experiment 
will  never  become  entirely  inaudible,  mainly  on  account 
of  the  transmission  of  sound  by  the  vibration  of  the  sup- 
porting wires,  we  are  given  reason  to  believe  that  with  com- 
plete exhaustion  and  the  removal  of  all  other  transmitting 
media,  no  sound  would  be  heard. 

165.  Velocity  of  Sound   Transmission. — Every  one   is 
familiar  with  the  fact  that  it  requires  time  for  sound  to 
travel  over  a  given  distance.     If  we  watch  a  locomotive 
from  a  distant  point  as  the  engineer  blows  the  whistle,  we 
first  observe  the  jet  of  steam  as  it  issues  from  the  whistle, 
and  a  few  seconds  later  we  hear  the  sound.     The  interval 
of  time  that  often  elapses  between  a  lightning  flash  and 
the  peal  of  thunder  is  further  evidence  that  the  velocity 
of  sound  is  not  exceedingly  great. 

Many  investigators  during  the  nineteenth  century  gave 
their  attention  to  the  accurate  determination  of  the  ve- 
locity of  sound.  For  the  most  part  their  experiments 
consisted  in  measuring  the  interval  of  time  between  the 
flash  of  a  gun  and  its  report  -heard  at  some  distant  sta- 
tion of  observation.  The  result  obtained  by  Regnault,  a 
French  physicist,  gives  sound  a  velocity,  in  air,  of  1085  feet 
per  second  at  the  temperature  of  freezing  water.  This  ve- 
locity is  equivalent  to  331  meters  per  second.  At  higher 
temperatures  the  velocity  is  somewhat  greater,  the  in- 
crease being  2  feet,  or  0.6  meter,  per  second  for  each  de- 
gree centigrade. 

166.  Velocity  of    Sound    in    Various   Mediums.  —  The 
velocity  of   sound  in  solids  has  been  the  subject  of  many 
investigations.     The   accepted   value   found   for   iron   is 
about  5100  meters  per  second  at  20°  C,     The  velocity  of 
sound  in  wood  depends  greatly  upon  the  kind  of  wood. 


SOUND:  ITS  NATURE   AND   PROPAGATION       169 

The  average  value,  however,  is  approximately  4000  meters 
per  second. 

The  most  exact  measurement  of  the  velocity  of  sound  in 
water  was  mads  in  1827  by  Colladon  and  Sturm  in  Lake 
Geneva,  Switzerland.  Two  observers  stationed  them- 
selves in  boats  at  opposite  sides  of  the  lake.  At  one  of 
the  stations  a  bell  was  sounded  beneath  the  water  and  a 
gun  fired  on  deck  at  precisely  the  same  instant.  The 
sound  was  received  by  means  of  a  large  ear  trumpet  held 
under  water  at  the  other  station.  The  time  intervening 
between  the  stroke  of  the  bell  and  the  report  transmitted 
by  the  water  could  thus  be  ascertained  and  the  velocity 
computed.  The  results  of  many  observations  gave  an 
average  of  1400  meters  per  second  as  the  velocity  of  sound 
through  water. 

EXERCISES 

1.  The  flash  of  a  gun  is  seen  3.5  seconds  before  the  report  is  heard. 
If  the  temperature  is  20°  C.,-what  is  the  distance  between  the  observer 
and  the  gun  ? 

2.  A  locomotive  whistle  was  sounded  3  mi.  from  an  observer.     If 
the  temperature  of  the  air  was  10D  C.,  how  long  was  the  sound  in 
traversing  the  distance  ? 

3.  The  distance  between  two  stations  is  12  mi.     If  the  interval  of 
time  between  the  flash  and  the  report  of  a  gun  was  found  by  experi- 
ment to  be  56  seconds,  what  was  the  speed  of  the  sound  ? 

4.  A  bullet  was  fired  at  a  target  500  in.  away,  and  in  3   seconds 
was  heard  by  the  gunner  to  strike.     The  temperature  of  the  air  being 
20°  C.,  what  was  the  velocity  of  the  bullet? 

5.  When  one  end  of  an  iron  pipe  is  struck  a  blow  with  a  hammer, 
an  observer  at  the  other  end  hears  two  sounds,  one  transmitted  by  the 
iron,  the  other  by  the  air.     If  the  pipe  is  1500  in.  long,  and  the  tem- 
perature 25°  C.,  what  is  the  interval  of  time  between  the  two  sounds  ? 

2.  NATURE  OF  SOUND 

167.  Sound  a  Wave  Motion.  —  We  have  seen  in  §  164 
that  a  medium  is  essential  for  the  transmission  of  sound 


170          A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

but  thus  far  no  explanation  has  been  given  of  the  manner 
in  which  this  transmission  takes  place.  Sounds  continue 
to  come  from  an  electric  bell  even  though  we  cover  it 
tightly  with  a  glass  jar,  but  it  is  very  plain  that  nothing, 
i.e.  no  material  thing,  can  pass  through  the  glass.  The 
whole  process  will  become  clear,  however,  if  we  consider 
that  a  sound  is  transmitted  through  the  air  and  glass  in  the 
form  of  waves.  Hence,  the  further  study  of  Sound  will  be 
a  study  of  waves  and  wave  motion. 

168.  Two  Kinds  of  Waves. — We  are  all  familiar  with 
the  waves  that  move  over  a  surface  of  water.  We  have 
only  to  observe  a  small  boat  as  it  rises  upon  the  crests  and 
sinks  down  into  the  troughs  to  realize  that  it  is  not  carried 
along  by  the  wave.  Again,  a  wave  is  often  seen  to  pass 
over  a  field  of  grain.  While  the  wave  moves  rapidly 
across  the  field,  each  spear  of  grain  simply  bends  with  the 
pressure  of  the  wind  and  then  rises  again.  The  following 
experiment  may  be  used  to  show  this  manner  of  wave 
propagation  : 

Let  one  end  of  a  soft  cotton  clothesline  about  25  feet  long  be 
attached  to  a  hook  in  the  wall,  while  the  other  end  is  held  in  the 
hand.  Give  the  end  of  the  rope  a  quick  up-and-down  motion,  and  a 
wave  will  be  seen  to  run  along  the  rope  from  one  end  to  the  other. 

In  the  experiment  just  described  it  is  clear  that  the  for- 
ward motion  of  the  waves  produced  in  the  rope  is  at  right 
angles  to  the  direction  of  the  motion  of  the  particles  compos- 
ing the  rope,  as  shown  in 
Fig.  132,  On  this  ac- 
count such  waves  are 
called  transverse  waves. 

FIG.  132.— Transverse  Waves  in  a  Rope.  A  second  kind  of  wave 

motion  takes  place  in  bodies  that  are  elastic  and  com- 
pressible, as  in  gases,  wire  springs,  etc.  We  are  able  to 
make  a  study  of  waves  of  this  kind  by  letting  a  coil  of 


SOUND:  ITS  NATURE  AND   PROPAGATION       171 

wire,  Fig.  133,  represent  the  medium  through  which  such 
waves  are  transmitted : 

Let  us  imagine  a  blow  is  given  the  spring  at  A  that 
quickly  compresses  a  few  turns  of  the  spiral  near  the  end. 
On  account  of  the  elasticity 
and  inertia  of  these  parts  of  A  .| 
the  spring,  they  will  move 

F,     6,.    .    /  FIG.  133.  — Waves  Transmitted  by  a 

forward     slightly    and     com-  Spring.      The     Individual    Turns 

press    those    just    ahead.  Move  back  and    forth  while   the 

*               .                  J  Waves  Progress  along  the  Spring. 

These  in  turn  will  compress 

the  coils  still  farther  along,  and  thus  a  pulse,  or  wave, 

is  carried  along  the  spring  from  A  to  R. 

Again  let  the  end  of  the  spring  at  A  be  given  a  very 
sudden  pull.  The  turns  near  the  end  will  be  drawn  apart 
for  an  instant,  but  the  adjacent  turns  will  be  drawn  toward 

A,  one  after  another,  until  the  end  B  receives  the  impulse. 
A  blow  against  A  is  transmitted  by  the  spring  to  the  point 

B,  and  a  sudden  pull  at  A  is  likewise  transmitted  as  a 
pull  against  the  fastening  at  B.     Waves  of  this  kind,  in 
which  the  motion  of  the  parts  of  the  medium  are  parallel 
to  the  direction  of   propagation,  are    called   longitudinal 
waves. 

169.  Transmission  of  a  Sound  Wave.  —  The  manner  in 
which  a  sound  is  transmitted  by  the  air  becomes  clear 
when  we  compare  the  process  with  that  which  occurs 
when  a  wave  is  transmitted  by  a  spiral  spring. 

Imagine  a  light  spring,  (1),  Fig.  134,  to  be  attached 
at  one  end  to  one  of  the  prongs  of  a  vibrating  tuning 
fork  F  and  at  the  opposite  end  to  a  diaphragm  G-;  Each 
vibration  of  the  fork  will  alternately  compress  and  separate 
the  spirals  of  the  spring  near  the  end.  These  pulses  will 
be  transmitted  by  the  spring  in  the  manner  described  in 
§  168,  and  will  cause  the  diaphragm  at  Gr  to  execute  as 
many  vibrations  per  second  as  the  tuning  fork,  and  the 


172 


A  HIGH  SCHOOL  COURSE  IN   PHYSICS 


diaphragm  will  give  out  a  sound  corresponding  to   that 
emitted  by  the  fork. 

Let  the  air  take  the  place  of  the  spring  and  the  ear  E 
replace  the  diaphragm.  When  the  prong  of  the  vibrating 
fork,  (2),  Fig.  134,  moves  to  the  right,  a  compression 
(>)  of  the  air  is  produced  in  front  of  it.  This  compression, 
or  condensation,  moves  to  the  right  with  the  velocity  of 
sound,  or  at  about  the  rate  of  about  1120  feet  per  second. 
When  the  prong  of  the  vibrating  fork  moves  to  the  left, 
the  air  just  at  the  right  is  rarefied,  and  the  adjacent  por- 
tions of  the  air  move  in  to  fill  this  rarefaction  (>),  which 

F  '  G\ 

-mmmmmmwmmmmmmmmmmmmmmmm 


(2) 


(3) 

FIG.  134.  —  Illustrating  the  Corresponding  Parts  of  Transverse 
and  Longitudinal  Waves. 

travels  to  the  right  immediately  following  the  condensa- 
tion. Condensation  and  rarefaction  thus  follow  one 
another  as  long  as  the  fork  continues  to  vibrate,  and  the 
drum  of  the  ear  at  E  receives  as  many  pulses  per  second 
as  the  tuning  fork  emits.  It  is  therefore  caused  to  vibrate 
as  many  times  per  second  as  the  fork.  Between  c  and  r  is 
w,  which  marks  the  region  where  there  is  neither  conden- 
sation nor  rarefaction. 

The  curve  in  (3),  Fig.  134,  is  drawn  to  show  the  parts  of 
a  transverse  wave  which  is  sometimes  used  to  represent 
diagrammatically  waves  of  other  kinds.  The  crest  AB 
corresponds  to  the  condensation,  or  compression,  of  the 
longitudinal  waves  shown  in  (1)  and  (2).  The  trough 


SOUND:  ITS  NATURE   AND   PROPAGATION       173, 

BC  corresponds  to  the  rarefaction.  The  distance  db  rep- 
resents the  distance  that  each  particle  in  the  wave  moves 
from  its  original  position  and  is  called  the  amplitude  of  the 
wave.  A  wave  includes  a  complete  crest  and  trough,  or  a 
condensation  and  a  rarefaction.  The  distance  between  the 
corresponding  points  on  any  two  adjacent  waves,  as  AC, 
BD,  etc.,  is  the  wave  length. 

170.  Velocity,  Wave  Length,  and  Vibration  Frequency.  — 
Imagine  a  tuning  fork  whose  rate  of  vibration,  or  vibration 
frequency,  is  256  per  second.     When  the  fork  is  set  in  vi- 
bration, it  sends  out  256  complete  longitudinal  waves  dur- 
ing each  second.     At  the  completion  of  the  256th  vibra- 
tion, the  first  wave  has  progressed  a  distance  numerically 
equal  to  the  velocity  of  sound,  or  about  1120  feet.     This 
space,  therefore,  contains  256  waves,  and  the  length  of  each 
wave  can  be  found  by  dividing  1120  feet  by  256.     Hence 
the  length  of  each  wave  is  about  4.4  feet.     Letting  n  be 
the  frequency,  v  the  velocity  of  sound,  and  I  the  wave 
length,  we  have 

V 

1  =  -,   or  nl  =  v.  (i\ 

n 

3.     INTENSITY   OF   SOUND 

171.  Sound  Waves  Spread  in  all  Directions.  — If  a  bell 
be  struck,  the  sound  is  heard  as  readily  in  one  direction 
as  another  if  no  ob- 
struction  intervenes 

and  other  conditions 
are  equally  favorable. 
Obviously  a  wave 

emitted    by    the  bell    FIG.  135.— SoundWaves  Spread  in  all  Directions 

spreads    out    in    all 

directions  from  it.     Figure  135  illustrates  this  fact.    Each 

condensation  has  the  form  of  a  hollow  spherical  shell  that 


174          A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

continually  enlarges  as  the  wave  advances.  This  is 
followed  by  a  rarefaction  of  a  similar  form.  It  is  plain 
that  the  energy  imparted  by  the  bell  to  a  single  wave  is 
carried  from  the  source  by  particles  of  air  composing  a 
hollow  spherical  shell.  At  a  given  distance  from  the  bell 
this  shell  will  have  a  certain  area  which  at  twice  the  dis- 
tance will  be  four  times  as  great,  since  the  area  of  a 
sphere  is  directly  as  the  square  of  its  radius.  Thus  at 
twice  the  distance  from  the  bell  the  energy  will 'be -im- 
parted to  four  times  as  many  particles,  hence  the  energy 
of  each  will  be  one  fourth  as  great.  At  three  times 
the  original  distance,  the  energy  will  be  imparted  to  nine 
times  as  many  particles,  and  the  energy  of  each  will  be 
one  ninth  as  much.  Hence,  the  intensity  of  sound  is  in- 
versely proportional  to  the  square  of  the  distance  measured 
from  the  source  of  the  sound. 

172.  Intensity  and  Amplitude.  —  The  energy  imparted 
to  each  wave  by  a  sounding  body  will  depend  upon  the 
intensity  of  the  vibration  of  that  body,  i.e.  upon  the  am- 
plitude of  vibration.  An  increase  in  the  amplitude  of 
vibration  of  the  sounding  body  produces  a  corresponding 
increase  in  the  amplitude  of  each  wave  emitted.  Thus 
each  wave  is  caused  to  carry  away  from  the  sounding 
body  an  increased  quantity  of  energy.  When  the  waves 
fall  upon  the  tympanum  of  the  ear,  a  corresponding  in- 
crease in  its  amplitude  of  vibration  is  produced. 

Let  one  of  the  prongs  of  a  tuning  fork  be  struck  lightly  against  a 
soft  pad  or  cork.  A  tone  is  emitted  that  is  scarcely  audible.  Now 
let  the  fork  be  struck  more  forcibly.  On  account  of  the  greater  am- 
plitude of  vibration  the  intensity  of  the  tone  will  be  much  greater 
than  before. 

Like  the  pendulum  (§§  79  and  80),  a  sounding  body 
completes  its  vibrations  in  equal  times,  the  period  of  vi- 
bration being  practically  independent  of  the  amplitude 


SOUND:  ITS  NATURE  AND  PROPAGATION   175. 

(i.e.  isochronous).  Therefore  when  the  fork  is  struck 
forcibly  against  the  pad,  it  still  performs  the  same  number 
of  vibrations  per  second  as  before. 

173.  Intensity  and  Density  of  the  Medium.  —  The  ex- 
periment with  the  bell  and  air  pump  (§  164)  shows  that 
the  intensity  of  a  sound  depends  upon  the  density  of  the 
medium  in  which  the  sound  is  produced.     As  the  air  in  the 
receiver  becomes  rarer,  the  intensity  of  the  sound  grows  less. 

Support  two  similar  bell  jars  with  the  mouths  opening  downward. 
Keep  one  filled  with  illuminating  gas  and  the  other  with  air.  Set  an 
electric  bell  ringing,  and  place  it  first  in  the  jar  containing  gas,  then 
in  the  one  filled  with  air.  It  will  be  readily  observed  that  the  sounds 
emitted  in  the  rarer  medium,  the  illuminating  gas,  are  less  intense 
than  those  produced  in  the  jar  filled  with  air. 

At  the  top  of  a  high  mountain  a  gun  makes  only  a 
small  report  when  fired.  At  high  elevations  explorers 
and  aeronauts  converse  with  difficulty.  The  denser  the 
gas,  the  greater  the  energy  imparted  to  each  wave  by  the 
vibrating  body.  For  this  reason  a  sounding  body  will 
cease  to  vibrate  sooner  in  the  denser  of  two  media. 

174.  Intensity  and  Area.  —  When  the  area  of  a  sound- 
ing body  is  small,  as  that  of  a  small  tuning   fork,  for 
example,  the  condensations  and  rarefactions  produced  are 
not  well  marked.     It  is  obvious  that  of   two  vibrating 
tuning  forks  of  the  same  frequency,  the  smaller  will  have 
the  effect  of  cutting  through  the  air  without  producing 
more  than  a  slight  compression,  while  the  one  of  larger 
area  will  compress  a  greater  volume  of  air,  and  thus  give 
out  more  of  its  energy  to  each  wave  emitted. 

Set  a  tuning  fork  in  vibration,  and  hold  the  shank  against  the 
panel  of  the  door  or  table.  A  loud  sound  will  be  heard  coming  from 
the  large  area  that  is  set  in  vibration  by  the  fork. 

In  the  construction  of  many  musical  instruments  use  is 
made  of  this  relation  between  the  intensity  of  sound  and 


176         A   HIGH   SCHOOL  COURSE   IN   PHYSICS 

the  area  of  the  sounding  body.  Thus  a  piano  utilizes  two 
or  three  wires  to  produce  a  given  tone  when  the  area  of 
a  single  wire  is  not  sufficiently  great.  Again,  a  sounding 
board  of  large  area  is  often  placed  beneath  the  strings  of 
instruments  to  form  a  large  surface  of  vibration  when  the 
strings  are  excited. 

4.    REFLECTION    OF   SOUND 

175.  Reflection  of  Sound  Waves.  — It  is  a  fact  frequently 
illustrated  in  nature  that  sound  waves  may  be  reflected. 
As  long  as  waves  proceeding  from  a  source  of  sound  pass 
through  a  homogeneous  medium,  no  reflection  takes 
place ;  but  when  the  density  of  the  medium  is  disturbed, 
the  waves  suffer  partial  or  total  reflection. 

Let  a  watch  be  placed  a  few 
inches  in  front  of  a  concave  re- 
flector, as  shown  in  Fig.  136.  By 
moving  the  ear  from  point  to  point 
a  place  may  be  found,  sometimes 
several  feet  from  the  reflector, 
where  the  sound  of  the  watch  may 
FIG.  136.  -  Sound  Reflected  by  a  Con-  be  distinctly  heard. 

cave  Surface.  .  .    , 

The  experiment  may  be  varied 

by  setting  a  similar  reflector  at  a  distance  of  several  feet  from  the 

first  and  facing  it,  as  shown  in  Fig.  137.     If  the  watch  is  placed 

slightly  nearer  the  reflector  than  in  the  preceding  experiment,  a  point 

may  be  found  a  few  inches  in  front 

of  the  second  at  which  the  sound 

is  focused.     This  point,  or  focus, 

may    be   found   by  using   an  ear 

trumpet  mipde  by  attaching  a  piece 

of  rubber  tubing  to  a  glass  funnel. 

When  the  open  funnel  is  placed  at  Fl«- 137.  — Sound  Undergoes  Two  Re- 

the  focus  of  sound,  a  distinct  tick-       flections  from  Concave  Surfaces" 

ing*  of  the  watch  will  be  heard  by  holding  the  end  of  the  rubber 

tubing  in  the  ear. 


SOUND:  ITS  NATURE   AND   PROPAGATION       177 

176.  Echoes.  —  The  familiar  phenomenon  of  echoes  is 
due  to  the  reflection  of  sound.  When  one  speaks  in  a 
room  of  moderate  size,  the  waves  reflected  from  the  walls 
reach  the  ear  so  quickly  that  they  combine  with  the  direct 
waves  and  produce  an  increase  of  the  intensity  of  the  sound. 
But  when  the  walls  are  one  hundred  feet  or  more  from 
the  speaker,  he  hears  a  distinct  echo  of  each  syllable  he 
utters.  Sometimes  a  reflecting  surface  is  so  distant  that 
several  seconds  may  intervene  before  an  echo  returns. 
Between  two  reflecting  surfaces  the  echoes  sent  back  and 
forth  are  often  remarkable.  It  is  related  that  at  a  point 
in  Oxford  County,  England,  an  echo  repeats  a  sound  from 
fifteen  to  twenty  times.  Extraordinary  echoes  are  also 
found  to  occur  between  the  walls  of  deep  canons. 

In  so-called  "  whispering  galleries  "  we  have  illustrated 
the  formation  of  a  sound  focus  due  to  curved  surfaces.  In 
the  crypt  of  the  Pantheon  in  Paris  there  is  a  place  where 
the  slight  clapping  of  hands  at  one  point  gives  rise  to 
sounds  of  great  intensity  at  another.  In  the  Mormon 
Tabernacle  at  Salt  Lake  City,  Utah,  a  whisper  near  one 
end  can  be  distinctly  heard  at  the  other,  so  perfect  is  the 
reflection  from  the  ellipsoidal  surfaces  of  the  walls. 

EXERCISES 

1.  A  hunter  fires  a  gun  and  hears  the  echo  in  5  seconds.    How  far 
away  is  the  reflecting  surface,  the  temperature  being  20°  C.? 

2.  How  far  is  a  person  from  the  wall  of  a  building,  if,  on  speaking 
a  syllable,  he  hears  the  echo  in  4  seconds,  the  temperature  being 
15°  C.? 

3.  A  gun  is  fired,  and  the  echo  is  returned  to  the  gunner  from  a 
cliff  250  ft.  away  in  4.5  seconds.     Calculate  the  velocity  pf ^ound. 

SUMMARY         y 

1.    Sounds  are  produced  by  bodies  in  a  state  of  rapid 
vibration  (§  161). 
13 


178          A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

2.  Sounding  bodies  vibrate  as  a  consequence 'of  the  elas- 
ticity and  inertia  of  the  material  composing  them  (§  162). 

3.  Sounds  are  transmitted  from  their  sources  by  solids, 
liquids,  and  gases.     The  air  is  the  propagating  medium  in 
most  cases  (§§  163  and  164). 

4.  The  velocity  of  sound  waves  in  air  at  the  freezing 
temperature  is  1085  ft.  (331  in.)  per  second.     The  velocity 
is  increased  2  ft.   (0.6  m.)  per  second  for   each   degree 
centigrade.     In  wood  and  iron  the  velocity  of  sound  is 
respectively  3  and  4  (approximately)  times  the  velocity 
in  air  (§§  165  and  166). 

5.  Sound   is  a  wave   motion.      Wave  motion  may  be 
either  transverse  or  longitudinal  (§§  167  and  168). 

6.  The  energy  of  a  sounding  body  is  transmitted  by 
longitudinal  waves.     The  parts  of  such  waves  are  called 
condensations  and  rarefactions.     The  corresponding  parts 
of  transverse  waves  are  crests  and  troughs  (§  169). 

7.  The   relation   between   velocity,   wave   length,  and 
number  of  vibrations  per  second  is  given  by  the  equation 
v  =  ln  (§170). 

8.  The  intensity  of  a  sound  depends  on  the  amplitude 
and  area  of  the  sounding  body,  the  density  of  the  medium 
where  the  sound  is  produced,  and  is  inversely  proportional 
to  the  square  of  the  distance  from  its  source  (§§  171  to  174). 

9.  Echoes  are  sounds  reflected  from  walls,  woods,  hills, 
cliffs,  etc.  (§§  175  and  176). 


CHAPTER  X 


SOUND :  WAVE   FREQUENCY  AND   WAVE   FORM 
1.  PITCH  OF  TONES 

177.  Musical    Sounds    and    Noises. — In   order   that   a 
sound  may  be  pleasing  to  the  ear,  it  is  essential  that  the 
vibrations  be  made  in  precisely  equal  intervals  of  time ; 
in  other  words,  the  vibrations  must  be  isochronous.     Any 
device  that  produces  isochronous  pulses  emits  a  musical 
sound.     If  the  pulses  are  not  isochronous,  the  result  is  a 
noise. 

Rotate  on  a  whirling  table  a  metal  or  cardboard  disk  provided  with 
two  or  more  circular  rows  of  holes,  Fig.  138.  Let  the  holes  in  one  row 
be  equidistant,  while  those  in  the  other  rows 
are  placed  at  irregular  intervals.  Blow  a 
stream  of  air  through  a  rubber  tube  against 
the  row  of  equidistant  holes,  and  a  pleasant 
musical  sound  will  result.  Now  direct  the 
stream  against  another  row  of  holes,  and 
it  will  be  observed  that  the  sound  produced 
is  of  an  unpleasant  character. 

178.  Pitch.  —  Probably  the    most 
striking  difference  in  musical  sounds 
is  in  respect  to  that  which  we  call 

.    FIG.  138.  —  Musical  Sounds 

pitch,  a  term  applied  to  the  degree  of  Produced  by  isochro- 
highness  or  lowness  of  a  sound.  We  nous  Pulses  in  the  Air- 
are  accustomed  to  sounds  varying  in  pitch  from  the  low, 
rumbling  thunder  to  the  shrill,  piercing  creak  of  small 
animals  and  insects.  How  this  wide  difference  in  sounds  is 
brought  about  may  be  shown  by  the  following  experiment: 

179 


180         A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

Rotate  a  metal  or  cardboard  disk  in  which  there  are  several  circu- 
lar rows  of  holes  perforated  at  regular  intervals,  but  having  a  different 
number  in  each  row.  See  Fig.  139.  Keeping  the 
speed  of  rotation  constant,  blow  a  stream  of  air 
forcibly  against  one  row  and  then  another.  A  dis- 
tinct difference  in  pitch  will  be  observed.  Again, 
keeping  the  stream  of  air  directed  against  one  of 
the  rows,  change  the  speed  of  rotation  from  very 

FIG     139 The     s^ow  to  verv  *as*'     ^e  pitch  will  rise  from  a  low 

Siren  Disk.          tone  to  one  that  is  very  shrill. 

These  facts  teach  that  the  pitch  of  a  tone  depends  upon 
the  number  of  wave  pulses  per  second  sent  from  a  sounding 
body  to  the  ear.  The  stream  of  air  blown  against  the  disk 
is  alternately  transmitted  and  interrupted  by  the  motion 
of  the  holes.  The  succession  of  pulses  thus  produced 
constitutes  the  musical  tone  that  is  heard.  The  vibration 
frequency  of  the  tone  emitted  may  be  found  by  multiply- 
ing the  number  of  revolutions  of  the  disk  per  second  by 
the  number  of  holes  in  the  circle.  An  instrument  of  the 
kind  used  in  the  experiment  is  called  a  siren. 

179.  Musical  Scales  Produced  by  a  Siren.  —  The  most 
common  series  of  tones  of  different  pitches  is  the  musical 
scale.  It  is  possible  with  the  help  of  a  siren  designed  in 
a  manner  similar  to  the  one  shown  in  Fig.  139  to  deter- 
mine the  relation  of  the  several  tones  that  comprise  the 
ordinary  musical  scale.  The  construction  and  use  of  the 
siren  is  as  follows : 

On  a  circular  disk  of  cardboard  or  metal,  about  10  inches  in  diame- 
ter, draw  eight  concentric  circles  about  £  inch  apart.  Upon  the  circles 
thus  drawn,  beginning  with  the  smallest,  drill  the  following  numbers 
of  equidistant  holes :  24,  27,  30,  32,  36,  40,  45,  and  48.  Mount  the 
disk  upon  a  whirling  table  and  rotate  with  a  uniform  speed.  Be- 
ginning with  the  smallest  circle,  blow  a  stream  of  air  against  each 
row  of  holes  in  succession.  The  tones  produced  will  be  recognized 
at  once  as  those  belonging  to  the  major  scale.  Increase  the  speed  of 
rotation,  and  again  direct  the  current  of  air  against  the  several  rows 


SOUND:  WAVE  FREQUENCY  AND  WAVE  FORM   181 

in  succession.     Although  the  pitches  are  higher  than  before,  yet  the 
scale  is  produced  as  perfectly  as  at  first. 

The  experiment  teaches  that  the  major  scale  is  a  series 
of  tones  whose  vibration  frequencies  have  the  same  rela- 
tion as  the  numbers  24,  27,  80,  32,  36,  40,  45,  and  48. 
To  these  tones  we  give  the  names  do,  re,  mi,  fa,  sol,  la, 
ti,  do.  These  tones  form  the  foundation  upon  which  has 
been  built  up  our  musical  system. 

180.  The  Major  Diatonic  Scale.  —  Any  series  of  tones 
whose  vibration  frequencies  bear  the  relations  given  in  the 
preceding  section  constitute  a  major  diatonic  scale.  The 
first  tone  of  such  a  series  is  the  key  tone.  The  various 
scales  in  use  are  named  according  to  their  key  tones ;  for 
example,  the  scale  of  C,  the  scale  of  Gr,  e<x).  Physicists  as- 
sign to  the  tone  called  middle  C,  256  vibrations  per  second.1 
Hence  the  second  tone  of  the  major  scale  of  C  must  be 
produced  by  256  x  f  J,  or  288  vibrations  per  second.  This 
tone  is  called  D.  The  next  tone,  or  E,  must  have 
256  X  |J,  or  320  vibrations  per  second,  etc.  The  follow- 
ing table,  Fig.  140,  shows  the  manner  in  which  these  tones  are 
expressed  on  the  musical  staff,  their  relative  and  absolute 
vibration  numbers,  and  the  vibration  ratios. 


Scale 
otC 

rJ  —                         —  ' 

Staff             if?T\ 

—  ^  — 

-&— 

—  ^  — 

ouiu                IvL/ 

o 

Letters 

T 

I 

1 

I 

1 

! 

i 

i' 

Absolute  Vib.  No. 

256 

288 

320  3kl.3 

38k 

k26.6 

k80 

51^ 

Tone  Number 

1 

2 

3 

k 

R 

g. 

7 

8 

For  all 

Syllables 

do 

re 

mi 

fa 

ftol 

la 

U 

do 

Scales 

Relative  Vib.  No. 

2k 

27 

30 

3fi 

ko 

kZ 

k8 

Vibration  Ratios 

1 

1 

§ 

§ 

¥ 

2 

Fia.  140.    The  Major  Scale. 

The  nature  of  the  seven  definite  intervals  leading  from 
C  to  0',  or  one  octave,  may  be  seen  from  the  table.     Start- 

1  The  international  standard  of  pitch  in  this  country  and  Europe  is 
based  upon  A  =  435  vibrations  per  second. 


182          A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

ing  with  C  —  256  vibrations,  the  vibration  numbers  of  the 
successive  tones  of  the  scale,  as  shown  by  the  experiment 
in  the  preceding  section,  bear  the  ratios  |,  f,  |,  |,  J, 
lf~,  and  2  to  the  vibration  number  of  O,  the  key  tone. 
These  ratios  are  the  same  for  all  major  scales,  no  matter 
what  the  key  tone  may  be. 

181.  Intervals.  — A  musical  interval  refers  to  the  relation 
between  the  pitches  of  two  tones.     The  experiment  with  the 
siren  in  §  179  shows  that  the  value  of  an  interval  depends 
upon  the  ratio,  not  the  difference,  between  the  vibration 
numbers  of  the  tones;    for  example,  when  this  ratio  is 
If,  or  2:1,  the  interval  is  an  octave,  as  the  interval  be- 
tween O'  and  C.     Other  important  intervals  are  the  sixth, 
i.e.  the  interval  between  the  first  tone  arid  the  sixth  of  a 
major  scale,  given  by  the  ratio  f  |,  or  5:3;    the  fifth,  |J, 
or  3:2;   the  fourth,  f  |,  or  4:3;    the  major  third,  f|-,  or 
5:4;  and  the  minor  third,  f  $,  or  6  :  5.    In  order  to  determine 
the  interval  between  two  tones,  the  ratio  of  their  vibration 
frequencies  is  first  computed  and  then  compared  with  the 
ratios  given  for  the  several  cases.     Hence  two  tones  of  500 
and  300  vibrations  per  second  respectively  are  a  sixth  apart, 
because  the  ratio  of  their  vibration  numbers  is  5:3.     If 
these  two  tones  were  sounded  together  or  in  succession, 
the  interval  would  be  recognized  at  once  by  a  musician. 

182.  The  Major  Chord.  —  A  careful  inspection  of   the 
vibration  numbers  of  the  tones  of  the  major   scale  will 
show  the  presence  of  three  groups  consisting  of  three  tones 
each,  whose  frequencies  bear  the  ratios  4:5:6.     Such  a 
group  of  tones  is  a  major  chord.     Beginning  with  C=  256 
vibrations  per  second,  we  have  for  the  first  chord  O,  E, 
and  Gr  (do-mi-sol)  whose  vibration  frequencies  are    256, 
320,  and  384.     This  chord  is  called  the  tonic  triad  of  the 
scale  of  O.     The  second  of  these  chords,  called  the  sub- 
dominant  triad,  is  formed  in  the  same  manner  and  includes 


SOUND:  WAVE  FREQUENCY  AND  WAVE  FORM   183 

F,  A,  and  0'  (fa-la-do).  The  third  is  formed  likewise 
by  making  use  of  the  tones  (r,  JP,  and  Dr  (sol-ti-re)  and 
is  called  the  dominant  triad.  Since  these  triads  include 
all  the  tones  of  the  major  scale,  it  may  be  said  that  this 
scale  is  founded  upon  these  three  major  chords.  The  fol- 
lowing table  shows  these  relations: 

Tonic  Triad  4  :  5  :  6  : :  256  :  320  :  384,  C,  £,  and  G. 

Subdominant  Triad    4  :  5  :  6  : :  341  :  427  :  512,  F,  A,  and  C>. 
Dominant  Triad         4  :  5  :  6  : :  384  :  480  :  576,  G,  B,  and  D'. 

183.  Sharps  and  Flats.  —  The  introduction  of  the  black 
keys  on  the  organ  or  piano  keyboard  is  (1)  for  the  pur- 
pose of  accommodating  the  instrument  to  the  range  of  the 
voice  in  the  case  of  songs  and  (2)  to  give  variety  to  selec- 
tions designed  for  instrumental  performance.  That  it.  is 
necessary  to  insert  additional  tones  becomes  apparent  at 
once  when  we  consider  the  tones  that  are  required  to  form 
a  major  scale  beginning  upon  Bv  just  below  middle  (?,  hav- 
ing 240  vibrations  per  second.  The  keys  that  are  used  in 
the  scale  of  O  are  all  white  keys,  Fig.  141,  and  have  the 
frequencies  indicated.  ^. 
In  the  major  scale  of  Eg 
B  the  .  vibration  fre- 
quencies must  be  suc- 
cessively 240,  270, 
300,  320,  360,  400,  450, 
and  480.  It  will  be 
observed  that  the  only  s  s  s  § 

white  keys  that  satisfy  FIG.  141.  —  Illustrating  the  Scale  of  C  on  Staff 
this  Scale  are  fl=  320  and  Keyboard. 

and  B  =  480  vibrations  per  second.  Since  the  number  270 
lies  about  midway  between  the  frequencies  of  0  and  .Z),  the 
black  key  (7*  (read  "(7 sharp'")  is  introduced.  Others  must 
be  placed  between  D  and  E,  F  and  6r,  G-  and  J.,  and  A  and 
B.  These  are  called  respectively  D*,  JF*,  6r*,  and  A*. 


184          A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

These  four  tones  are  also  called    JS^    6r^,  A  ,  and  B  , 
and  are  read  «E  flat,"  "  G  flat,"  etc. 

184.  Tempered  Scales.  —  In  the  preceding  section  is 
shown  the  necessity  for  introducing  additional  tones  in 
order  that  a  piano,  for  example,  may  be  used  to  produce 
the  major  scale  of  B.  These  new  vibration  numbers,  how- 
ever, will  not  satisfy  scales  which  begin  on  other  key 
tones,  for  every  change  to  another  key  tone  increases  the 
demand  for  small  changes  in  the  vibration  numbers  of  the 
tones.  The  difficulty  is  surmounted  by  sacrificing  the 
perfect,  simple  ratios  found  in  §§  180  to  183  and  substi- 
tuting others  that  are  sufficiently  near  to  satisfy  a  musical 
ear.  See  the  table  given  below.  This  method  of  tuning 
an  instrument  is  called  tempering,  and  the  scales  derived 
by  the  process  are  called  tempered  scales.  The  desired 
result,  which  is  to  permit  the  execution  of  musical  compo- 
sitions written  in  any  key,  is  secured  by  making  the  inter- 
val between  any  two  adjacent  tones  the  same  throughout  the 
length  of  the  keyboard.  By  this  process  the  octave  is  di- 
vided into  twelve  precisely  equal  intervals  called  half 
steps.  The  imperfection  introduced  by  equal  tempera- 
ment tuning  is  illustrated  by  the  following  table : 

CDEFGABC 

Perfect  scale  of  C  256.0  288.0   320.0  341.3   384.0  426.6  480.0  512.0 
Tempered  scale      256.0  287.3   322.5   341.7  383.6  430.5  483.3   512.0 

EXERCISES 

1.  Taking  G  as  the  key  tone  of  a  major  scale,  compute  the  vibra- 
tion frequency  of  each  of  the  tones  contained  in  an  octave. 

2.  The  vibration  frequency  of  a  tone  is  264.     Calculate  the  fre- 
quencies of  its  third,  fourth,  and  octave. 

3.  Calculate  the  vibration  frequency  of  the  tone  one  octave  below 
middle  C. 

4.  What  is  the  wave  length  of  a  tone  whose  vibration  frequency  is 
256  when  the  temperature  is  15°  C.  ?     (See  §§  165  and  170.) 


SOUND:  WAVE   FREQUENCY  AND   WAVE  FORM      185 

5.  If  middle  C  were  given  the  frequency  260,  what  would  be  the 
frequencies  of  D,  A ,  and  B  ? 

6.  A  tone  two  octaves  above  middle  C  has  how  many  vibrations 
per  second  ? 

7.  If  the  keyboard  of  a  piano  extends  three  and  a  half  octaves  in 
each  direction  from  middle  C,  calculate  the  vibration  number  of  the 
lowest  and  the  highest  C  on  the  instrument. 

8.  Ascertain  the  interval  between  the  pitches  of  two  tuning  forks 
making  256  and  192  vibrations  per  second. 

9.  The  tones  of  three  bells  form  a  major  chord.     One  makes  200 
vibrations  per  second,  and  its  pitch  lies  between  the  pitches  of  the 
others.     Calculate  the  frequency  of  the  bells. 

10.  A  tone  is  produced  by  a  siren  revolving  300  times  per  minute. 
What  is  the  name  of  the  tone  produced,  if  the  number  of  holes  in  the 
row  is  24  ? 

2.   RESONANCE 

185.  Sympathetic  Vibrations.  —  Vibrations  that  are  pro- 
duced in  one  body  by  another  near  by  which  has  the  same 
vibration  period  are  called  sympathetic  vibrations.  The 
following  experiments  will  serve  to  illustrate  the  case  : 

1.  Tune  two  wires  of  a  sonometer,  Fig.  150,  so  that  they  emit  tones 
of  the  same  pitch  when  plucked.     Place  a  A-shaped  paper  rider  astride 
one  of  the  wires  and  then  pluck  the  other.     The  rider  will  be  thrown 
off,  and  if  the  wire  that  was  plucked  be  stopped,  a  tone  will  be  heard 
coming  from  the  other.     Throw  the  wires  slightly  out  of  unison  and 
repeat  the  experiment.     Only  very  slight  vibrations  will  be  imparted 
to  the  second  wire. 

2.  Let  two  mounted  tuning  forks  having  the  same  pitch  be  placed 
near  together,   as  shown  in  Fig. 

142.  Set  one  of  the  forks  in  vi- 
bration by  bowing  it,  or  by  strik- 
ing it  with  a  rubber  stopper 
attached  to  a  wooden  or  glass 
handle.  On  checking  its  vibra- 
tions a  sound  will  be  heard  com-  FlG.  142.  —  Sympathetic  Vibrations 
ing  from  the  second  fork.  between  Forks  in  Unison. 

Students  will  find  it  interesting  to  experiment  in  a  simi- 
lar manner  with  a  piano.  For  example,  let  the  key  C  be 
depressed  so  carefully  that  no  tone  is  produced.  The 


186 


A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


damper  is  thus  lifted  from  the  wires  of  a  certain  vibration 
frequency.  Now  if  the  tone  O  be  sounded  by  a  voice,  the 
corresponding  tone  will  be  heard  coming  from  the  instru- 
ment. 

These  experiments  show  the  facility  with  which  a  body 
is  put  in  vibration  by  another  having  the  same  frequency. 
The  principle  may  be  stated  as  follows: 

Sound  waves  sent  out  from  one  vibrating  body  impart  vi- 
brations to  others,  provided  the  vibration  frequencies  are  the 
same. 

186.  Sympathetic  Vibrations  Explained.  —  The  phenom- 
enon of  sympathetic  vibrations  is  readily  accounted  for 
when  we  consider  that  the  body  first  set  in  vibration  sends 
out  periodic  impulses  through  the  air  or  the  wood  connect- 
ing the  two  bodies.  The  first  impulse  received  by  the 
second  body  gives  it  a  very  small  amplitude  of  vibration. 
It  returns  at  precisely  the  instant  to  receive  the  second 
impulse,  and  the  amplitude  of  the  second  vibration  is  caused 

to  be  greater  than  that  of  the  first. 
The  third  impulse,  in  like  manner, 
adds  energy  to  that  already  trans- 
mitted to  the  body.  Thus,  after 
the  arrival  of  several  impulses  the 
amplitude  of  vibration  of  the  sec- 
ond body  is  great  enough  for  the 
production  of  an  audible  tone. 
187.  Resonance.  —  A  vibrating 


able  Length. 


FIG.  143.  —  Recn- 
forcement  of  a 
Sound  by  an  Air 

Column  of  Suit-  timing  fork  when  held  in  the  hand 
produces  a  sound  that  is  scarcely 
audible.  Its  tone  may  be  aug- 
mented, however,  by  placing  it 
above  a  properly  adjusted  «ir  column  that  is  caused  to  vi- 
brate sympathetically  with  it. 


SOUND:  WAVE  FREQUENCY  AND  WAVE  FORM   187 

Let  a  vibrating  timing  fork  be  held  over  a  tall  cylindrical  jar,  as 
shown  in  Fig.  143.  By  pouring  water  into  the  jar  while  the  fork  is  in 
this  position,  a  condition  will  be  reached  such  that  an  intense  sound 
may  be  heard  proceeding  from  the  jar.  Pouring  in  more  water  de- 
stroys the  effect. 

Let  the  experiment  be  repeated  with  a  fork  of  a  different  pitch. 
If  the  pitch  is  higher,  the  air  column  will  need  to  be  shortened  by 
pouring  in  more  water ;  if  lower,  the  column  must  be  longer. 

188.  Resonance  Explained.  — Let  a  and  5,  Fig.  144,  rep- 
resent the  two  extreme  positions  of  the  prong  of  a  vibrat- 
ing tuning  fork.     Just  as  the  prong  begins     a^ 

a  downward  swing  from  a  it  starts  a  conden-       — '"'"lirrs*' 
satioii  downward  in  the  tube.     When   the 
prong  begins  its  upward  swing  from   5,  it 
starts   a   condensation  in  the  air  above  it. 
Now,  in  order  to  have  a  reenforcement  of 
the  tone  emitted,  the  condensation  started 
downward  in  the  tube  must  be  reflected  by 
the  water  and  return  in  time  to  unite  with   FIG    144  _The 
the  condensation  produced  above  the  prong       Cause  of  Reso- 
as  it  moves  upward.     Hence  the  condensa- 
tion must  travel  down  the  tube  and  back  while  the  prong 
of  the  fork  moves  from  a  to  6,  i.e.  during  one  half  a  vibra- 
tion of  the  fork.     In  a   similar   manner    the    rarefaction 
started  in  the  tube  as  the  prong  leaves  5  must  return  from 
the  bottom  of  the  tube  in  time  to  unite  with  the  rarefaction 
produced  above  the  prong. 

189.  Resonance  Tubes  and  Wave  Length.  — "In  the  ex- 
planation of  the  phenomenon  of  resonance  given  in  the 
preceding  section,  it  is  clear  that  the  waves  in  the  air  of 
the  resonance  tube  travel  twice  the  length  of  the  air  space 
within  the  tube  while  the  fork  is  making  one  half  a  vibra- 
tion.     The  wave  therefore  goes   four   times   the  length 
of  the  air  column  during  a  complete  vibration  of  the  fork. 
But  a  wave  progressesj^jlyvave  length  during  a  complete 


'Bjjjk™ 


188          A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

vibration  of  a  sounding  body.  Hence  the  length  of  the 
vibrating  air  column  is  one  fourth  of  the  length  of  the  air 
wave  produced  by  the  fork. 

Experiments  performed  with  tubes  of  different  sizes 
have  shown  thlH  •diameter  of  the  tube  influences  the 
length  of  tube  ^^^Hld  for  the  best  effect.  It  has  been 
found  that  the  length  of  the  air  column  must  be  increased 
by  two  fifths  of  its  diameter  in  order  to  equal  a  quarter  of 
a  wave  length. 

Tuning  forks  are  often  mounted  on  properly  adjusted 
boxes  whose  air  spaces  act  as  resonators  when  the  forks 
are  sounded,  as  shown  in  Fig.  142. 

3.    WAVE  INTERFERENCE  AND  BEATS 

190.  Destruction  of  One  Wave  by  Another.  —  Let  us 
imagine  two  trains  of  sound  waves,  (1)  and  (2),  Fig.  145. 

Let  the  waves  in  the 
two  trains  be  of-  the 
same  length,  i.e.  pro- 
duced by  sounding 

145. -Two  Longitudinal   Waves  in  the     b()dies    Qf    th        game 
Condition  Necessary  for  Interference. 

pitch.  •  If  the  con- 
densations in  one  train  at  a  given  instant  are  at  <?,  c  and 
those  in  the  other  train  at  c(,  c',  it  is  obvious  that  the  two 
trains  cannot  move  through  the  same  air  simultaneously. 
For,  if  the  compressions  are  equal,  the  condensations  of 
the  first  train  will  unite  with  the  rarefactions  r',  r'  of  the 
second,  and  the  result  will  be  no  change  in  the  density  of 
the  air.  In  other  words,  one  train  of  waves  will  destroy 
the  other.  Thus  two  sounds  may  combine  and  produce 
silence.  A  study  of  the  analogous  case  of  transverse 
waves  will  help  to  make  the  matter  clearer. 

Let  (1)  and  (2),  Fig.  146,  represent  two  trains  of  trans- 
verse waves.  Let  their  lengths  and  amplitudes  be  equal. 


SOUND:  WAVE   FREQUENCY  AND   WAVE   FORM       189 


FIG.  146.  — The  perference  of  Trans- 
verse Waves. 


If,  now,  the  crests  <?,  c,  e  of  one  train  unite  with  the 
troughs  £,  £,  t  of  the  other,  then  one  train  will  obviously 
destroy  the  other,  since 

t 


the  crests  of  the  first 
train  will  just  fill  the 
troughs  of  the  second 
and  vice  versa.  Thus 
two  trains  of  water  waves, 
for  example,  may  combine  and  produce  a  level  sur- 
face. 

The  destruction  of  one  wave  train  ~by  another  similar  train 
is  called  interference. 

191.    An  Example  of  Interference.  —  Let /,/,  Fig.  147, 
represent  the  ends  of  the  two  prongs  of  a  tuning  fork.    At 

the  moment  the  prongs  vibrate 
toward  each  other  a  condensa- 
tion is  produced  in  the  region 
a  and  rarefactions  at  b,  b. 
When  the  prongs  move  out- 
ward, a  rarefaction  is  produced 
at  a  and  condensations  in  the  re- 
gions b,  b.  Now  since  a  conden- 
sation starts  from  the  region  a 
at  the  instant  a  rarefaction  starts 
from  5,  and  vice  versa,  there  will 
be  places  in  the  air  around  the  tuning  fork  where  the  parts 
of  the  waves  coming  from  a  will  unite  with  the  unlike 
parts  of  those  from  6,  and  continuous  interference  will 
result.  The  regions  of  interference  near  the  prongs  of 
a  fork  are  shown  by  the  dotted  lines  in  the  figure.  These 
places  may  be  found  by  rotating  a  tuning  fork  held  near 
the  ear.  Positions  in  which  the  sound  becomes  almost 
inaudible  can  thus  be  easily  located. 


Fia.  147.— Regions  of  Silence 
near  a  Tuning  Fork. 


190          A   HIGH    SCHOOL  COURSE   IN    PHYSICS 

Hold  a  vibrating  tuning  fork  above  a  resonance  tube  tuned  to  re- 
enforce  the  tone  emitted.  Rotate  the  fork  slowly,  and  a  position  will 
be  found,  Fig.  148,  in  which  the  sound  be- 
comes practically  inaudible;  for,  in  this 
position,  the  mouth  of  the  jar  receives  the 
unlike  parts  of  waves  sent  out  from  the 
opposite  sides  of  the  nearer  prong  of  the 
fork. 

192.  Alternate  Interference  and 
FIG  148  —An  in-  Reenforccment.  —  When  it  is  under- 
terference  Phe-  stood  how  two  trains  of  sound  waves 
may  combine  and  produce  an  inten- 
sified sound  (§  188)  or  interfere  and 
produce  silence  (§  190)  the  inter- 
esting phenomenon  of  beats  is  read- 
ily explained.  Beats  are  always  produced  when  two  tones 
that  differ  slightly  in  pitch  are  sounded  at  the  same  time. 
The  effect  is  obtained  as  follows : 

Tune  two  resonance  jars  to  reenforce  two  tuning  forks  of  the  same 
pitch.  Load  the  prongs  of  one  of  the  forks  with  pieces  of  tin  bent  in 
such  a  form  as  to  cling  firmly  to  the  fork  while  it  is  in  vibration. 
This  will  make  the  pitch  of  this  fork  slightly  lower  than  that  of  the 
other.  Now  sound  the  two  forks  simultaneously  and  hold  them  over 
the  resonance  tubes.  Fluctuations  in  the  intensity  of  the  sound  (i.e. 
beats")  will  be  distinguishable  at  a  distance  of  several  meters.  Load 
the  prongs  of  the  fork  more  heavily,  and  more  rapid  beats  will  be 
observed. 

Since  the  forks  used  in  the  experiment  differ  in  pitch, 
the  waves  sent  out  will  differ  in  length,  —  the  longer 
waves  being  produced  by  the  fork  with  the  weighted 
prongs.  Since  the  pitches  differ  but  a  small  amount,  the 
wave  lengths  will  be  almost  equal.  Inasmuch  as  the 
vibrations  of  the  weighted  fork  are  continually  falling 
behind  those  of  the  other,  at  certain  periods  the  two  forks 
will  be  producing  condensations  and  rarefactions  simul- 


SOUND:  WAVE  FREQUENCY  AND  WAVE  FORM   191 

taneously ;  hence  reinforcement  results.  A  little  later 
the  weighted  fork  will  have  fallen  one  half  a  vibration 
behind  the  other,  and  while  one  fork  is  producing  a  con- 
densation, the  other  will  be  sending  out  a  rarefaction. 
Hence,  at  this  instant,  there  is  interference  between  the 
two  trains  of  waves. 

Figure  149  shows  portions  of  the  two  trains  of  waves 
sent  out  from  the  forks.     It  will  be  observed  that  at  r,  r 


*A\S~^7     ^      ^~  ~^ — V/    \J 

FIG.  149.  —  The  Production  of  Beats  Illustrated. 


the  two  trains  unite  to  produce  the  greatest  reinforcement 
of  the  sound,  while  at  i  unlike  portions  of  the  waves  unite 
and  thus  destroy  the  sound.  The  resultant  wave  is  repre- 
sented by  the  line  AB. 

193.  Law  of  Beats.  —  Let  one  tuning  fork  make  100 
vibrations  per  second,  and  another,  101.  At  a  given 
instant  imagine  each  fork  to  be  sending  out  a  condensa- 
tion. Hence,  at  this  instant,  reenforcernent  takes  place. 
Just  one  half  of  a  second  later,  one  fork  has  made  50  com- 
plete vibrations,  and  the  other  50J.  Therefore,  at  this 
instant  one  fork  is  producing  a  rarefaction,  and  the  other 
a  condensation,  and  thus  the  resulting  sound  is  weakened. 
Hence,  the  intensity  of  the  sound  will  increase  and  de- 
crease once  per  second.  This  effect  constitutes  one  beat. 
When  the  difference  in  vibration  frequency  is  two,  the 
phenomenon  occurs  twice  per  second.  In  every  case  the 
number  of  beats  produced  per  second  is  equal  to  the  differ- 
ence between  the  vibration  frequencies  of  the  two  sounding 
bodies. 


192          A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


4.    THE  VIBRATION  OF  STRINGS 

194.  The  Pitch  of  Strings. — Many  musical  instruments 
employ  vibrating  strings  or  wires  on  account  of  the  fact 
that  such  bodies  emit  tones  of  a  rich  quality.     The  wide 
range  of  pitch  necessary  is  obtained  by  varying  the  length, 
tension,  and  mass  of  the  strings  used.     Thus  the  lengths 
of  the  wires  of  a  piano,  for  example,  vary  from  those  that 
are  about  2  inches  long  to  others  whose  lengths  are  4.  or 
5  feet.     The  short  wires  are  of  a  small  diameter,  while  the 
long  ones  are  large  and  massive.     The  violin  makes  use 
of  four  strings  of  different  masses,  and  the  performer  ob- 
tains the  necessary  pitches  by  fingering  the  strings  so  as 
to  allow  the  proper  length  to  be  put  in  vibration. 

195.  Law  of  Length.  —  The  laws  governing  the  vibration 
of  strings  may  be  shown  by  means  of  an  instrument  called 


FIG.  150.  —  A  Sonometer. 

the  sonometer  (pronounced  so  nom'e  ter),  Fig.  150.  The 
relation  between  the  pitch  of  a  string  and  its  length  may 
be  shown  by  the  following  experiment : 

Adjust  the  tensions  of  the  two  similar  wires  of  a  sonometer  until 
they  emit  tones  of  the  same  pitch.  Set  up  a  bridge  A  and  place  a 
second  bridge  under  the  middle  of  one  of  the  wires  and  pluck  first 
one  wire  and  then  the  other.  It  will  be  observed  that  the  tones  pro- 
duced are  an  octave  apart;  i.e.  one  half  the  original  length  of  the 
wire  produces  a  pitch  of  twice  (§  180)  as  many  vibrations  per  second. 
Another  test  may  be  made  by  placing  a  bridge  at  C  just  two  thirds 
of  the  distance  AB  from  A.  By  plucking  both  wires  as  before  the 
interval  do-sol  is  easily  recognized. 


SOUND:  WAVE  FREQUENCY  AND  WAVE  FORM   193 

This  experiment  may  be  extended  to  the  production  of 
any  interval  that  can  be  recognized.  In  any  case  it  will 
be  found  that  the  ratio  of  the  length  of  the  vibrating 
portion  AC  to  the  original  length  AB  is  the  inverse  of 
the  vibration  ratio  given  in  §  180.  Thus  one  half  the 
length  produces  a  pitch  of  twice  as  many  vibrations  per 
second,  and  two  thirds  of  the  original  length  emits  a  tone 
of  three  halves  of  the  vibration  frequency.  Hence,  the 
vibration  frequencies  of  strings  or  wires  are  inversely  pro- 
portional to  their  lengths. 

196.  Law  of  Tensions.  — Let  the  two  weights  used  to  produce 
the  tensions  in  the  wires  of  a  sonometer  be  moved  as  far  out  as  possible 
on  the  lever  arms.     Set  the  bridges  so  that  the  wires  emit  tones  of  the 
same  pitch.     Now  move  one  weight  back,  thus  shortening  the  lever 
arm  (§  93)  until  it  produces  only  one  fourth  as  much  tension  on  this 
wire  a»  before.     The  pitch  will  now  be  found  to  be  an  octave  lower 
when  the  two  wires  are  plucked. 

This  experiment  may  be  applied  to  other  intervals ;  for 
example,  four  ninths  of  the  original  tension  will  cause  the 
vibration  frequencies  to  have  the  ratio  of  2  :  3.  Hence, 
the  vibration  frequencies  of  strings  or  wires  are  directly 
proportional  to  the  square  roots  of  their  tensions. 

197.  Law  of  Masses.  —  Let  a  sonometer  be  equipped  with  two 
wires  of  the  same  material,  lengths,  and  tensions,  but  of  different  sizes. 
If  possible,  make  the  diameter  of  one  twice  that  of  the  other.     Since 
the  masses  are  as  the  squares  of  the  diameters,  the  mass  of  a  given 
length  of  the  larger  will  be  four  times  that  of  the  smaller.     If  now 
the  wires  are  plucked,  the  pitch  of   the  smaller  will  be  an  octave 
above  that  of  the  other. 

The  vibration  frequencies  of  strings  or  wires  are  inversely 
proportional  to  the  square  roots  of  their  masses  per  unit  length. 

198.  Vibration  of  a  String  in  Parts.  —  The  vibration  of 
a  stretched  string  or  wire  is  more  complicated  than  at  first 

appears.     When  a  string  is  bowed  or  plucked,  the  tone 
14 


194 


A  HIGH   SCHOOL  COURSE  IN   PHYSICS 


emitted  is  a  compound  tone  produced  by  the  string 
vibrating  as  a  whole  simultaneously  with  its  vibration  in 
parts.  The  readiness  with  which  a  string  vibrates  in  parts 
may  be  shown  by  the  following  method : 

Attach  one  end  of  a  white  silk  cord  about  1  m.  in  length  to  one  of 
the  prongs  of  a  large  tuning  fork  attached  firmly  to  the  table  and 
whose  frequency  is  not  more  than  100.  Let  the  other  end  pass  over 


FIG.  151.  —  A  Cord  Vibrating  as  a  Single  Segment. 


a  smooth  hook  or  pulley  and  support  a  weight,  as  in  Fig.  151.  Set 
the  fork  in  vibration  and  adjust  the  tension  and  length  of  tfce  cord 
until  it  vibrates  as  a  single  segment,  as  shown.  Now  reduce  the  weight 

FIG.  152.  — The  Vibration  of  a  Cord  in  Two  Parts. 

used  until  the  cord  vibrates  in  two  parts,  Fig.  152,  when  the  fork  is 
set  in  motion.  Under  suitable  tensions  the  cord  will  vibrate  in  any 
desired  number  of  equal  parts  up  to  six  or  seven. 

The  tone  emitted  when  a  string  vibrates  as  a  whole  is  called 
its  fundamental.  The  fundamental  is  the  lowest  tone  that 
a  string  can  produce.  By  the  pitch  of  a  string  is  meant 
the  pitch  of  its  fundamental.  The  vibrating  portions  of 
the  string  are  called  loops,  or  segments,  and  a  point  where 
the  amplitude  of  vibration  is  zero  is  called  a  node.  Loops 
are  often  called  ventral  segments  and  antinodes. 

199.  Overtones  of  Strings.  —  The  character  of  the  tone 
produced  by  a  vibrating  string  is  complicated  by  the  divi- 
sion of  the  string  into  equal  parts,  as  shown  in  the  preced- 
ing section.  Each  vibrating  segment  of  a  string  produces 


SOUND:  WAVE  FREQUENCY  AND  WAVE  FORM   195 

a  tone  that  is  higher  in  pitch  than  the  fundamental.  The 
tones  emitted  by  the  vibrating  portions  of  any  sounding  body 
are  called  overtones,  or  partial  tones. 

The  presence  of  overtones  emitted  when  a  wire  is 
plucked  may  be  detected  by  the  sympathetic  vibrations 
set  up  in  a  neighboring  wire. 

Let  the  two  wires  of  a  sonometer  be  tuned  in  unison.  Place  a 
bridge  under  the  center  of  one  wire  and  set  A-shaped  paper  riders 
near  the  middle  of  each  half  of  this  wire.  If  the  longer  wire  is  now 
plucked  near  one  end,  the  shorter  wires  will  be  thrown  into  vibration, 
as  the  riders  will  show.  Replace  the  riders,  and  pluck  the  longer 
wire  at  the  center.  Only  feeble  vibrations  are  now  set  up  in  the 
shorter  wires.  Again,  place  the  bridge  under  one  wire  just  one  third 
of  the  original  length  from  one  end,  and  place  a  rider  astride  the 
shorter  portion.  Now  if  we  pluck  the  longer  wire  near  one  end  as 
before,  the  rider  will  be  thrown  off  as  in  the  former  case.  Replace  the 
rider  and  pluck  the  longer  wire  one  third  of  its  length  from  one  end. 
The  rider  will  remain  stationary. 

In  this  experiment  we  have  used  the  shorter  wire  to 
show  when  certain  overtones  are  present  in  the  tone  emitted 
by  the  longer  wire.  The  results  indicate  that  when  a 
wire  is  plucked  near  one  end,  the  tone  produced  contains 
an  overtone  an  octave  higher  than  the  fundamental.  The 
wire  not  only  vibrates 
as  a  whole,  but  divides 

into      two      parts,      each        FIG.  153.- A  Wire  Emitting  itsFunda- 
part  having  a  frequency  mental  and  First  Overtone. 

double  that  of  the  fundamental.  This  condition  is  illus- 
trated in  Fig.  153.  When  the  wire  is  plucked  near  the 
center,  this  overtone  is  not  present,  since  a  node  would  be 
required  at  this  point. 

200.  A  Series  of  Overtones.  —  The  second  part  of  the 
experiment  in  the  preceding  section  shows  that  the  wire 
divides  into  three  equal  parts,  each  of  which  vibrates  with 
three  times  the  frequency  of  the  fundamental.  In  a  simi- 


196          A   HIGH   SCHOOL  COURSE   IN   PHYSICS 

lar  manner  it  is  possible  to  detect  the  presence  of  still 
higher  overtones,  all  of  which  are  produced  simultaneously 

OF 


5th  overtone,  .  X  256  vibs. 

d11  ov";  J  S  H  3ft     near  one  end.     The  following  table 

2d   overtone,  3  x  256  vibs.  ,  ,  .    .  ,    ,, 

overtone,  2  x  256  vibs.     shows  the  positions  and  frequencies 
.  256  vib..     of  the  overtones  produced  when  a 

middle    O  string   is   plucked   near 

FIG.   154.—  Table  Showing 

the  Overtones  Produced      one  end. 

on  a  Middle  c  string.  When  the  vibration  frequency  of 

an  overtone  is  2,  3,  4,  5,  etc.,  times  that  of  the  fundamen- 
tal, it  is  called  an  harmonic.  The  relative  intensity  of  the 
various  overtones  produced  by  a  wire  depends  chiefly  up- 
on the  manner  in  which  it  is  set  in  motion,  the  point  where 
it  is  struck  or  plucked,  and  the  rigidity,  density,  and  elas- 
ticity of  the  wire. 

5.  QUALITY   OF    SOUNDS 

201.  Overtones  and  Tone  Quality.  —  It  is  a  familiar  fact 
that  two  tones  of  the  same  pitch  and  intensity  do  not  neces- 
sarily sound  alike.  The  tones  produced  by  a  violin,  for 
example,  are  readily  distinguished  from  those  of  the  piano 
or  flute.  The  tones  of  one  violin  may  differ  from  those  of 
another.  This  difference  is  one  of  quality.  The  cause  of 
this  difference  between  tones  was  long  a  matter  of  study 
and  investigation.  It  was  finally  explained  fully  by  the 
German  physicist  Helmholtz1  (1821-1894).  He  tells  us 
that  the  quality  of  a  sound  depends  upon  the  overtones 
produced  "by  the  sounding  body  and  their  relative  intensities. 

Let  a  wire  be  plucked  at  one  end  and  the  tone  compared  with  that 
emitted  when  the  wire  is  plucked  in  the  middle.  Again,  let  the  wire 
be  struck  with  a  soft  rubber  hammer  and  then  with  something  hard. 
The  tones  produced  will  be  of  the  same  pitch  but  of  different  quality. 
This  last  experiment  may  also  be  made  with  either  a  bell  or  a  tuning 
fork. 

1  See  portrait  facing  page  196. 


HERMANN    VON    HELMHOLTZ    (1821-1894) 

A  new  era  in  the  history  of  Sound  was  created  in  1862  by  the 
publication  of  Helmholtz's  Lehre  von  den  Tonempfindungen,  a 
work  which  has  been  translated  into  English  under  the  title  Sensa- 
tions of  Tones.  The  author  recognizes  musical  tones  as  periodic 
motions  of  the  air  and  distinguishes  the  three  characteristics  of 
tones  as  intensity,  pitch,  and  quality.  He  finds  that  tone  quality 
is  due  to  the  number  and  relative  intensity  of  the  upper  partials,  or 
overtones.  Helmholtz  devised  hollow  spherical  resonators  of  a  vari- 
ety of  sizes  by  the  aid  of  which  he  was  able  to  analyze  the  human 
voice  and  other  musical  tones.  By  means  of  a  large  series  of 
tuning  forks  of  different  pitches  which  were  operated  electrically, 
he  produced  tones  of  such  a  composition  as  to  imitate  the  vowel 
sounds  of  the  human  voice,  the  tones  of  organ  pipes,  etc. 

Helmholtz  received  a  medical  education  at  Berlin,  and  from 
1855  to  1871  was  professor  of  physiology  at  Bonn  and  Heidelberg. 
He  was  led  to  the  study  of  physics  in  his  endeavor  to  understand 
the  principles  involved  in  the  eye  and  ear.  Later  he  became  an 
accomplished  mathematician.  In  1871  Helmholtz  was  appointed 
professor  of  phyics  at  Berlin,  and  in  1888  became  the  first  director 
of  the  well-known  Reichsanstalt  (Imperial  Physico-Technical  Insti- 
tute) in  Charlottenburg.  His  death  occurred  in  1894. 

The  first  contribution  of  Helmholtz  to  the  science  of  physics  is 
his  famous  treatise  on  the  Conservation  of  Energy  published  in 
1847,  a  publication  which  was  of  great  influence  in  establishing 
this  doctrine.  He  is  known  throughout  the  medical  world  as  the 
inventor  of  the  ophthalmoscope,  an  instrument  used  in  examining 
the  interior  of  the  eye. 


SOUND:  WAVE  FREQUENCY  AND  WAVE  FORM   197 

These  experiments  teach  that  the  quality  of  the  sound 
produced  by  a  body  depends  upon  the  manner  in  which  the 
body  is  set  in  motion.  As  shown  in  §  199,  the  sound  given 
out  by  a  wire  is  rich  in  overtones  when  plucked  near  one 
end;  but  when  plucked  at  the  center,  all  the  overtones 
that  require  a  node  at  that  point  are  absent.  When  a 
tuning  fork  is  struck  with  a  hard  substance,  high  ringing 
overtones  are  plainly  heard,  while  the  fundamental  is  ex- 
tremely weak.  A  very  different  quality  is  produced,  how- 
ever, by  bowing  the  fork  or  striking  it  against  a  soft  pad. 

202.  Stringed  Instruments.  —  Among  the  most  common  musi- 
cal instruments  which  depend  on  the  vibration  of  strings  and  wires  are 
the  piano,  violin,  guitar,  mandolin,  and  banjo.  The  piano  consists  of 
a  series  of  tightly  stretched  wires,  which  in  the  lower  octaves  are  very 
massive  and  vary  in  length  from  three  to  five  feet.  The  shortest  wires 
of  the  instrument  are  often  not  more  than  two  inches  in  length.  When 
a  key  is  struck,  the  motion  is  transmitted  through  a  system  of  levers  to 
a  padded  hammer  which  strikes  the  wires  tuned  to  give  the  correspond- 
ing pitch.  The  tone  emitted  is  greatly  intensified  by  the  vibrations 
produced  in  the  sounding-board.  When  the  key  is  released,  a  padded 
damper  falls  against  the  vibrating  wires,  thus  checking  the  motion. 
A  pressure  of  the  foot  upon  the  right-hand  pedal  of  the  instrument 
raises  all  the  dampers  and  thus  leaves  the  vibration  of  every  wire 
unchecked.  Changes  of  temperature  and  humidity  of  the  atmosphere 
gradually  put  the  wires  out  of  tune.  The  necessary  tuning  is  accom- 
plished by  a  suitable  adjustment  of  the  tension  of  each  wire. 

Instruments  of  the  violin  type  which  are  played  with  a  bow,  as  well 
as  the  guitar,  mandolin,  and  banjo,  derive  their  musical  tones  from 
properly  tuned  wires  or  strings  whose  lengths  may  be  changed  at  will 
by  the  performer.  The  violinist  by  long  practice  learns  the  exact 
position  at  which  to  press  the  strings  against  the  finger  board  to  pro- 
duce the  desired  tones.  By  this  method  any  pitch  from  that  of  the 
lowest  string  to  the  one  derived  from  the  shortest  possible  portion  of 
the  string  of  highest  fundamental  can  be  produced.  The  difficulty  is 
reduced  in  the  case  of  the  guitar,  mandolin,  and  banjo  by  the  slightly 
raised  frets,  or  bridges,  placed  across  the  finger  board  to  indicate  the 
position  of  the  finger.  These  instruments  owe  their  wide  difference 
in  tone  quality  (1)  to  the  nature  of  the  string,  (2)  to  the  manner  in 


198          A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

which  the  strings  are  put  in  vibration,  and  (3)  to  the  size,  shape,  and 
material  of  their  sounding-boards. 

EXERCISES 

1.  How  are  the  different  pitches  produced  which  are  necessary  in 
rendering  a  selection  on  a  guitar,  harp,  mandolin,  or  cello? 

2.  The  bridges  under  a  stretched  wire  are  4  ft.  apart.     Where  must 
a  third  bridge  be  placed  to  raise  the  pitch  a  major  third?  a  minor 
third? 

SUGGESTION.  —  See  §  181  for  the  vibration  ratios  for  these  intervals. 

3.  A  wire  180  cm.  long  produces  middle  C.     Show  by  a  diagram 
where  a  bridge  would  have  to  be  placed  to  cause  the  string  to  emit 
each  tone  of  the  major  scale  of  C. 

4.  What  is  the  vibration  frequency  of  the  tone  three  octaves  above 
middle  C?    What  length  of  the  wire  in  Exer.  3  would  be  required 
to  produce  this  tone? 

5.  The  tension  of  a  string  is  9  kg.     What  tension  must  it  have  in 
order  to  produce  a  tone  an  octave  higher?  an  octave  lower?  a  fifth 
higher? 

6.  Write  the  vibration  frequencies  of  the  tones  one,  two,  and  three 
octaves  both  above  and  below  middle  C. 

7.  Write  the  vibration  frequencies  of  the  first  four  overtones  of  a 
G  string.     Name  these  tones. 

8.  The  density  of  steel  is  7.8,  and  that  of  brass  8.7.     What  is  the 
vibration  frequency  of  a  steel  wire,  if  that  of  a  brass  wire  of  equal 
length,  diameter,  and  tension  is  280  ? 

SUGGESTION.  —  Let  x  be  the  vibration  frequency  of  the  steel  wire 
and  then  write  the  proportion  based  on  the  law  of  masses,  §  197. 

9.  The  vibration  frequency  of  two  equal  wires  155  cm.  long  is  300. 
How  many  beats  per  second  will  be  heard  when  one  of  the  wires  has 
been  shortened  5  cm.? 

10.  Two  middle  C  forks  were  placed  near  together  and  the  prongs 
of  one  of  them  weighted  with  bits  of  sealing  wax ;  when  both  forks 
were  sounded,  4  beats  per  second  were  heard.  Find  the  frequency  of 
the  weighted  fork. 

6.    VIBRATING  AIR  COLUMNS 

203.  Organ  Pipes. — The  pipe  organ,  flute,  clarinet, 
cornet,  etc.,  employ  resonant  air  columns  for  the  produc- 
tion of  musical  tones.  The  pitch  of  the  tones  emitted  by 


SOUND:  WAVE  FREQUENCY  AND  WAVE  FORM   199 

such  columns  depends  mainly  upon  the  length  of  the 
column.  The  relation  between  the  length  of  a  column 
and  its  pitch  is  shown  by  the  following  experiments: 

1.  Measure  the  length  of  various  organ  pipes  that  are  available  for 
experimental  purposes.     Compute  the  ratio  of  the  length  of  each  pipe 
to  the  length  of  pipe  giving  the  key  tone  of  a  major  scale.     This  ratio 
will  be  found  to  be  the  inverse  of  the  vibration  ratio  for  the  corre- 
sponding tone  of  the  scale  as  given  in  §  180. 

2.  Prepare  a  set  of  glass  or  metal  tubes  about  1  centimeter  in  diam- 
eter of  the  lengths  10,  15,  20,  and  30  centimeters.     Leaving  the  tubes 
open  at  both  ends,  blow  across  one  end  of  the  tube  10  centimeters 
long  in  such  a  manner  as  to  produce  an  audible  tone.     Even  if  the 
sound  is  not  loud,  its  pitch  can  usually  be  recognized  and  sung  by 
members  of  the  class.     Again,  blow  across  the  end  of  the  20-centi- 
meter tube,  and  compare  its  pitch  with  that  of  the  first  tube.     The 
interval  between  the  two  tones  will  be  practically  an  octave,  the 
shorter  tube  having  the  higher  pitch.     Verify  this  relation  by  using 
the  tubes  whose  lengths  are  15  and  30  centimeters.     In  a  similar 
manner  ascertain  the  interval  between  the  pitches  of  the  tones  emitted 
by  the  tubes  whose  lengths  are  20  and  30  centimeters.     This  interval 
will  be  about  a  fifth. 

Organ  pipes  are  either  open  or  stopped.  A  stopped  pipe 
is  formed  by  closing  one  end  of  a  tube.  The  experiment 
teaches  that  when  the  length  ratio  of  two  open  pipes  is  1:2, 
the  vibration  ratio  is  2 : 1 ;  and  when  the  former  ratio  is  2 :  3, 
the  latter  is  3:2.  Hence,  the  vibration  frequencies  of  open 
pipes  are  inversely  proportional  to  their  lengths. 

204.  Open  and  Stopped  Pipes.  —  Close  an  end  of  one  of  the  tubes 
used  in  Experiment  2  of  the  preceding  section,  and  cause  it  to  emit  its 
tone  as  before.  Produce  also  the  tone  of  the  pipe  when  both  ends  are 
open,  and  compare  the  pitches.  The  pitch  of  the  closed  tube  is  an  oc- 
tave lower  than  that  of  the  open  one.  This  result  may  be  verified  by 
using  any  of  the  tubes.  If  an  organ  pipe  is  available,  blow  it  first 
while  open  and  then  while  closed  tightly  at  one  end.  The  best  re- 
sults are  obtained  by  blowing  it  moderately. 

The  pitch  of  a  closed  pipe  is  an  octave  lower  than  that  of 
an  open  one  of  the  same  length. 


200 


A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


205.    Mechanism  of  an  Organ  Pipe.  —  Sectional  views  of 
open  organ  pipes  made  of  metal  and  wood  are  shown  in 

Fig.  155.  Air  is  blown 
from  a  wind  chest  into 
the  small  chamber  c  and 
flows  in  a  thin  stream 
through  the  narrow  ori- 
fice i  against  the  lip  a. 
At  a  the  air  is  set  in  vi- 
bration, and  this  in  turn 
puts  the  air  contained  in 
the  tube  in  vibration. 
Since  the  pitch  of  the 
pipe  is  determined  by  its 
length,  it  can  reenforce 
only  those  vibrations  at 
the  lip  a,  whose  vibration 
frequency  corresponds  to 
its  own. 
The  tube 
may  be  re- 
garded as  a 
resonator 
for  reenf  orc- 
A  vibrating 


FIG.  155.  — Open  Organ  Pipes.     (1)   A 
Metal  Pipe.     (2)   A  Wooden  Pipe. 


ing  a  tone  of  a  particular  pitch, 
tuning  fork  whose  pitch  is  the  same  as  that 
of  the  tube  would  also  set  the  air  column  in 
vibration  when  held  near  the  open  end. 

206.    Nodes  in   Vibrating   Air   Columns.  - 
When  an  open  pipe  is  yielding  its  fundamen- 
tal, or  lowest  tone,  a  node  n,  (1),  Fig.  156,  is 
formed  at  the  center.     The  arrows  indicate 
that  the  vibratory  motion  of  the  air  on  opposite  sides  of 
a  nodal  point  is  always  in  opposite  directions.     At  the 


FIG.  150.  — 
Nodes  in 
Open  and 
Stopped 
Pipes. 


SOUND:  WAVE  FREQUENCY  AND  WAVE  FORM  201 

open  ends  of  the  tube  aa  the  air  is  free  to  vibrate 
and  the  density  of  the  air  remains  practically  un- 
changed; hence  these  points  are  often  called  antinodes  or 
loops.  Since  the  distance  from  a  node  to  an  antinode  is  al- 
ways equal  to  a  quarter  of  a  wave  length  (§  189),  the  length 
of  the  entire  wave  is  four  times  na,  or  twice  the  length 
of  the  tube. 

The  case  of  stopped  pipes  is  different.  A  node  is  al- 
ways formed  at  the  closed  end  of  the  tube  n,  (2),  Fig.  156, 
and  an  antinode  near  a.  The  wave  length  is  four  times 
the  distance  an,  or  four  times  the  length  of  the  tube.  It 
thus  becomes  clear  why  the  fundamental  of  a  closed  pipe 
is  an  octave  below  that  of  the  open  one. 

207.  Overtones  Produced  by  Organ  Pipes.  —  By  blowing 
an  organ  pipe  forcibly,  tones  of  a  higher  pitch  than  the 

fundamental  will  be  produced. 

a        n        a        n        a 

This  is  also  true  of  the   tubes  (i)       i     ?    2         a    |    4 
used  in   Experiment    2,    §  203.         a     n     a     n     a     n     a 
Just  as  in  the  case  of  vibrating  (2)     '   !  2 — L_I_J — 5   !  6 

Strings    (§198),    the    vibrating      FIG.  157.  —  Overtones  of  Open 

body,  which  is  the  air  column  in 

this  instance,  divides  into  parts  according  to  existing 
conditions.  It  is  by  the  vibration  of  these  parts  that  the 
higher  tones,  or  overtones,  are  produced.  Figure  157 
shows  the  position  of  the  nodes  and  antinodes  when  the 
pipe  is  sounding  its  first  and  second  overtones.  In  (1)  it  is 
shown  that  the  tube  contains  four  quarter  waves ;  hence 
the  wave  length  is  equal  to  the  length  of  the  tube.  Simi- 
larly for  the  second  overtone  (2),  the  division  of  the  vi- 
brating air  column  forms  six  quarter  waves.  Now  since 
the  vibration  frequency  increases  as  the  wave  length  de- 
creases, the  first  overtone  has  twice  the  frequency  of  the 
fundamental ;  the  second,  three  times  the  frequency  ;  and 
so  on.  Thus  the  overtones  of  an  open  pipe  are  the  same 


202 


A  HIGH   SCHOOL  COURSE  IN   PHYSICS 


as  those  of  vibrating  strings  (§  200),  and  are  therefore 
harmonics. 

The  division  of  the  air  column  in  stopped  pipes  is  other- 
wise. As  we  have  already  seen,  a  node  is  always  formed 
at  the  closed  end  and  an  antinode  at  the  open  end.  Hence, 
in  the  production  of  the  fundamental  tone,  only  one  quarter 
wave  is  contained  in  the  pipe.  But  when  the  first  over- 
tone is  emitted,  a  node  must  again  be  formed  at  the  closed 
end  w,  and  an  antinode  at  the  open  end.  Evidently,  if 
only  one  other  node  is  to  exist  it  must  be  at  n'  (1),  Fig. 

158,  one  third  of  the  length  of 
the  pipe  from  the  open  end. 
The  pipe  then  contains  three 
quarter  waves.  Similarly, 
when  yielding  its  second  and 
third  overtones,  the  column 
must  divide  into  five  and  seven  quarter  waves  respectively 
in  order  to  retain  a  node  at  the  closed  end  and  an  antinode 
at  the  open  end.  This  is  shown  for  the  second  overtone 
in  (2).  Hence  in  a  stopped  organ  pipe  only  those  over- 
tones can  be  produced  whose  frequencies  are  odd  multiples 
of  the  vibration  frequency  of  the  fundamental. 

208.  Wind  Instruments.  —  Under  this  title  are  classed  all  in- 
struments whose  tones  are  emitted  by  air  columns.  The  most  impor- 


(0 


(2) 


3  :  4 


FIG.  158.  — Overtones  of  Closed 
Pipes. 


(2) 


FIG.  159.— (1)  The  Flute.     (2)   The  Clarinet. 


tant  instrument  of  this  kind  is  tine  pipe  organ.  The  wide  range  of  pitch 
is  derived  from  organ  pipes  (see  Fig.  155)  of  different  lengths.  A 
great  variety  in  the  quality  of  tones  is  secured  by  the  use  of  pipes  of 


SOUND:  WAVE  FREQUENCY  AND  WAVE  FORM  203 


FIG.  160.  —  Mouthpiece  of  the 
Clarinet. 


different  kinds,  as  open  and  closed  wooden  pipes,  metal  pipes,  reed 
pipes,  etc. 

The  flute,  (1),  Fig.  159,  corresponds  to  an  open  organ  pipe  whose 
length  can  be  changed  by  the  aid  of 
openings  along  the  side  controlled  by 
the  fingers.  The  air  in  the  instru- 
ment is  set  in  vibration  by  blowing  a 
current  from  the  lips  forcibly  across 
the  opening  at  a. 

The  clarinet,  (2),  Fig.  159,  is  pro- 
vided with  a  mouthpiece,  Fig.  160,  which  contains  a  reed  or  tongue  of 
light,  flexible  wood,  which  alternately  opens  and  closes  the  aperture. 
The  action  of  the  instrument  resembles  that  of  the  so-called  "  squawker," 
made  of  the  stem  of  the  dandelion  or  by  cutting  a  tongue  on  the  side 
of  a  quill.  The  current  of  air  blown  by  the  lungs  causes  the  reed 
to  close  the  opening  momentarily.  There  is  thus  started  through 
the  tube  a  wave,  the  returning  reflected  pulse  from  which  forces  the 
reed  outward.  A  series  of  rapid  puffs  is  in  this  manner  maintained 
at  the  end  of  the  tube.  The  performer  secures  the  various  pitches 
by  openings  in  the  side  of  the  instrument,  as  in  the  case  of  the  flute. 
The  cornet.  Fig.  161_  is  provided  with  a  mouthpiece  of  the  form 

shown  in  Fig.  162,  within 
the  large  opening  of 
which  the  lips  are  caused 
to  vibrate.  The  neces- 


FIG.  161.  — The  Cornet. 


FIG.  162.  —  Mouthpiece 
of  the  Cornet. 


sary  range  of  pitch  is  obtained  by  blowing  overtones  and  controlling 
the  length  of  the  tube  by  the  valves.  (See  Fig.  154.)  The  trombone, 
Fig.  163,  is  an  instrument  of  the  cornet  type,  in  which  the  pitches  are 


c 


JUL 


FIG.  163.  —  The  Slide  Trombone. 


204          A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

produced  by  sliding  the  portion  ab  to  the  proper  positions  for  chang- 
ing the  length  of  the  vibrating  air  column. 

209.  Vibrating  Reeds  and  Diaphragms.  —  Reed  organs,  ac- 
cordions, and  mouth  organs  are  provided  with  reeds  similar  to  that 
shown  in  Fig.  164.  A  current  of  air  blown  in  the  direction  of  the 

arrow  suffices  to  set  the  tongue  a 
in  vibration.  Different  pitches 
are  obtained  by  making  the  reed 
of  suitable  length  and  rigidity. 

A    vibrating     disk,     or    dia- 
phragm, is  employed  in  the  tele- 
Fia.l64.-AnOrgaoReed.  phone    an(J     phonograph.     The 

description  of  the  electric  telephone  will  be  deferred  until  a  study 
has  been  made  of  its  underlying  principles.  The  mechanical  telephone 
may  be  made  by  simply  connecting  two  metal  diaphragms  with  a 
strong  cord  or  wire.  The  vibrations  set  up  at  one  end  by  speaking 
against  the  diaphragm  are  transmitted  by  longitudinal  waves  in  the 
wire  which  set  up  corresponding  vibrations  at  the  second  instrument. 
Such  telephones  are  used  for  short  distances  only. 

The  phonograph  affords  an  interesting  application  of  the  recording 
and  reproduction  of  sounds  by  the  aid  of  a  small  diaphragm.  In  the 
process  of  making  a  "  record  "  a  smooth,  or  blank,  cylinder  of  soft  ma- 
terial, as  wax,  is  placed  on  the  rotating  shaft  of  the  instrument,  and 
against  its  surface  is  carefully  adjusted  a  sharp  cutting  point  attached 
to  the  back  of  the  diaphragm.  The  tones  of  a  voice  or  instrument 
produce  sound  waves  which  are  collected  by  the  horn  and  transmitted 
to  the  diaphragm.  When  the  diaphragm  vibrates,  its  up-and-down 
motion  causes  the  point  to  engrave  in  the  surface  of  the  rotating  cyl- 
inder a  spiral  groove  of  ever- varying  depth  corresponding  to  the  com- 
plex form  of  the  sound  waves  which  fall  upon  it.  In  the  reproduction 
of  sound  the  record  is  mounted  on  the  revolving  shaft,  and  a  delicate 
stylus  which  is  attached  to  the  back  of  the  diaphragm  is  adjusted 
in  the  spiral  groove  on  the  surface  of  the  cylinder.  When  the  cylinder 
rotates,  the  stylus  rises  and  falls  with  the  little  irregularities  of  the 
groove  and  thus  sets  the  diaphragm  in  vibration.  Since  the  manner 
in  which  the  diaphragm  vibrates  is  governed  wholly  by  the  nature  of 
the  engraved  record,  the  sounds  which  it  emits  resemble  very  closely 
those  by  which  the  record  was  produced.  In  many  instruments  of 
this  type,  the  records  are  engraved  on  disks  instead  of  cylinders. 


SOUND:  WAVE   FREQUENCY   AND  WAVE   FORM     205 

EXERCISES 

1.  What  is  the  wave  length  of  the  tone  produced  by  an  open  pipe 
2  ft.  long?  by  a  closed  pipe  of  the  same  length? 

2.  Compute  the  length  of  a  middle  C  open  pipe,  the  temperature 
being  20°  C. 

SUGGESTION.  —  See  §  170  for  the  relation  of  wave  length  and  pitch. 

3.  A  whistle  may  be  regarded  as  a  stopped  pipe.     If  the  cavity  of 
a  whistle  is  1  in.  long,  find  the  vibration  frequency  of  its  tone  when 
the  temperature  of  the  air  with  which  it  is  blown  is  25°  C. 

4.  By  blowing  across  the  end  of  a  tube  6  in.  long,  closed  at  one 
end,  the  first  overtone  is  emitted.     Find  its  frequency,  the  tempera- 
ture being  18°  C. 

5.  Find  the  vibration  frequencies  of  the  first  four  overtones  of  a 
middle  C  pipe,  (1)  when  the  pipe  is  open  at  both  ends  and  (2)  when  it 
is  stopped  at  one  end. 

6.  When  a  small  stream  of  water  is  allowed  to  run  into  a  bottle,  a 
sound  is  heard.     Does  the  pitch  of  the  tone  rise  or  fall  ?     Explain. 

7.  If  the  ear  is  held  near  the  mouth  of  a  tall  jar  or  the  open  end  of 
a  tube,  a  tone  will  be  heard.     Perform  the  experiment,  and  then  ex- 
plain how  the  so-called  "sound  of  the  sea"  is  heard  coming  from 
large  sea-shells. 

SUMMARY 

1.  A  musical  sound  is  distinguished  from  a  noise  by  the 
isochronism  of  the  vibrations.     Pitch  is  governed  by  the 
number  of  vibrations  per  second  (§§  177  and  178). 

2.  A  major  diatonic  scale  is  a  series  of  eight  tones  whose 
vibration  numbers  are  to  each  other  in  the  relation  of  the 
numbers  24,  27,  30,  32,  36,  40,  45,  and  48,  or  which  bear 
the  following  ratios  to  the  first,  or  key  tone:  1,  -§-,  j,  £,  f, 
f,  Jys  and  2  (§§  179  and  180). 

3.  An  interval  is  the  relation  between  the  pitches  of 
two  tones.     It  depends  entirely  upon  the  ratio  of  their 
vibration  numbers.     The  common  intervals  are  the  octave, 
sixth,  fifth,  fourth,    major   third,   and   minor   third.     The 
intervals  are  expressed  by  definite,  simple  ratios  (§  181). 


206          A  HIGH   SCHOOL  COURSE   IN    PHYSICS 

4.  A  major  chord  consists  of  three  tones  whose  vibration 
numbers  are  as  4  :  5  :  6  (§  182). 

5.  Sharps  and  flats  are  used  for  the  purpose  of  supply- 
ing the  necessary  tones  for 'scales  other  than  the  scale  of  0. 
Scales  are  tempered  in  order  that  an  instrument  with  fixed 
tones  may  produce  all  scales  equally  well  (§§  183  and  184). 

6.  Resonance  depends  upon  the  fact  that  a  vibrating 
body  will  impart  vibrations  to   another  near  by,  whose 
natural  vibration  frequency  is  the  same  as  its  own  (§§  185 
to  189). 

7.  Two  trains  of  waves  will  weaken  or  destroy  each 
other  if  the  condensations  in  one  train  coincide  with  the 
rarefactions  of  the  other.     The  cancellation  of  sound  is 
complete   when   the   amplitudes    and   wave    lengths    are 
equal  (§§  190  and  191). 

8.  The  coincidence  of  two  trains  of  waves  which  differ 
slightly  in  length  results  in  the  alternate  reenforcement 
and  weakening  of  the  resultant  tone.     The  fluctuations  of 
intensity  produced  in  this  manner  are  called  beats.     The 
number  of  beats  per  second  is  equal  to  the  difference  of 
the  vibration  numbers  of  the  two  tones  (§§  192  and  193). 

9.  The  vibration  frequency  of  a  stretched  string  or 
wire   depends   on   its   length,    mass,    and   tension.      The 
frequency  is  inversely  proportional  to  the  length  and  the 
square  root  of  the  mass  per  unit  length  and  directly  pro- 
portional to  the  square  root  of  the  tension  (§§  194  to  197). 

10.  A  string  as  a  rule  vibrates  as  a  whole  and  at  the 
same  time  in  parts.     The  tone  produced  by  the  vibration 
as  a  whole  is  its  fundamental,  and  the  tones  emitted  by 
the  vibrating  parts  are  its  overtones  (§§  198  and  199). 

11.  Since  a  string   divides  into   equal  parts,  i.e.   into 
halves,  thirds,  quarters,  etc.,  the  vibration  numbers  of 
the  overtones  are  2,  3,  4,  5,  etc.,  times  the  frequency  of  the 


SOUND:  WAVE  FREQUENCY  AND  WAVE  FORM  207 

fundamental.      Such     overtones     are    called     harmonics 
(§  200). 

12.  The  quality  of  a  sound  is  dependent  on  the  over- 
tones present  and  their  relative  intensities  (§  201). 

13.  The  pitch  of  an  organ  pipe  depends  upon  its  length. 
The  vibration  numbers  of  the  fundamentals  emitted  by 
pipes  are  inversely  proportional  to  their  lengths.     The 
fundamental  of  a  stopped  pipe  is  an  octave  lower  than 
that  of  an  open  pipe  of  equal  length  (§§203  to  205). 

14.  The  wave  length  of  the  fundamental  of  an  open 
pipe  is  twice,  and  of  a  stopped  pipe  four  times,  the  length 
of  its  air  column  (§  206). 

15.  The  overtones  of  open  pipes  are  all  the  harmonics, 
as  is  the  case  for  strings ;  but  the  overtones  of  stopped  pipes 
are  only  those  that  have  respectively.  3,  5,  7,  etc.,  times 
the  vibration  frequency  of  the  fundamental  (§  207). 


CHAPTER  XI 

HEAT:  TEMPERATURE  CHANGES  AND  HEAT 
MEASUREMENT 

1.  TEMPERATURE  AND  ITS  MEASUREMENT 

210.  Temperature. — Among  the  most  common  experi- 
ences of  everyday  life  are  the  sensations  of  warmth  and 
coldness  as  we  come  near  or  in  contact  with  objects  around 
us.  These  sensations  enable  us  to  distinguish  between 
different  bodies  with  respect  to  that  condition  called 
temperature.  If  several  vessels  of  water  or  pieces  of  iron, 
for  example,  are  placed  before  us,  we  are  able  by  the  sense 
of  feeling  to  arrange  them  in  the  order  of  their  various 
temperatures.  On  this  account,  we  find  in  common  lan- 
guage many  terms  made  use  of  to  express  what  may  be 
called  the  degree  of  hotness  of  bodies ;  as  hot,  warm, 
tepid,  lukewarm,  cool,  and  cold.  We  say  that  one  body 
is  warmer  than  another;  or,  to  use  another  expression,  has 
a  higher  temperature  than  the  other. 

Although  our  temperature  sense  is  of  great  value  to  us 
at  all  times,  it  does  not  afford  an  infallible  guide  in  every 
instance,  as  the  following  experiments  will  show : 

1.  Place  the  hand  against  the  wooden  portion  of  the  class-room 
seat,  and  then  transfer  it  to  the  iron  part.     State  which  feels  the 
warmer. 

2.  Let  three  vessels  of  water  be  prepared :    the  first  very  warm, 
the  second  very  cold,  and  the  third  lukewarm.     Place  the  right  hand 
in  the  cold  water  and  the  left  hand  in  the  warm  water,  and  hold  them 
there  about  a  minute.     Now  transfer  the  left  hand  to  the  third  vessel. 
The  water  will  feel  cold.     Now  remove  the  right  hand  from  the  cold 
water,  and  place  it  in  the  third  vessel.     The  same  water  feels  warm. 

208 


UNIVERSITY 

OF 

'"TIRE   CHANGES   AND  MEASUREMENT     209 

In  the  first  experiment  the  temperatures  of  the  iron 
and  wood  are  practically  the  same ;  but  the  hand  which 
is  warmer  than  either  the  wood  or  iron  falls  most  rapidly 
in  temperature  when  in  contact  with  the  iron  than  when 
touching  the  wood.  Furthermore,  it  is  a  well-known  fact 
that  a  room  may  be  considered  warm  by  a  person  who  has 
been  running,  and  cold  by  another  who  has  been  sitting 
quietly  in  the  house. 

211.  Relation  between  Temperature  and  Heat.  — When  a 
body  which  has  a  certain  temperature  becomes  hotter,  we 
ascribe  the  cause  of  this  change  to  the  acquisition  of  heat. 
We  say  that  hea.t  has  been  added  to  it.  When  such  a 
body  becomes  colder,  we  say  that  heat  has  been  taken 
from  it.  The  rise  in  the  temperature  of  a  body  may 
be  produced  in  many  different  ways,  e.g.  an  iron  is  heated 
in  a  fire,  a  piece  of  steel  by  its  friction  with  a  moving 
grindstone,  an  electric  lamp  by  a  current  of  electricity,  or 
the  earth  by  the  rays  of  the  sun. 

It  is  necessary  for  us  to  distinguish  carefully  between 
the  temperature  of  a  body  and  its  heat.  The  fact  is  well 
known  that  a  cup  of  water  taken  from  a  boiling  kettle  is 
just  as  hot  as  the  water  remaining  in  the  kettle ;  or 
a  pail  of  water  dipped  from  a  pool  has  at  that  instant 
precisely  the  same  temperature  as  the  pool.  But  it  is 
also  very  evident  that  of  two  hot-water  bottles  filled 
from  the  same  kettle,  the  one  containing  the  greater 
quantity  of  water  will  give  out  more  heat  and  for  a 
longer  time  than  the  smaller  bottle.  Hence,  while  the 
temperatures  of  two  bodies  may  be  equal,  the  amounts 
of  heat  contained  within  them  may  be  very  different. 
Thus  a  thermometer  placed  in  a  vessel  of  water,  for 
example,  indicates  the  temperature  of  the  water,  but  in 
no  way  does  it  show  the  amount  of  heat  which  the  water 

contains. 

15 


210         A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

212.  Nature  of  Heat.  —  In  order  to  investigate  the 
nature  of  that  which  brings  about  the  changes  in  the  tem- 
peratures of  bodies,  let  the  following  experiments  be  per- 
formed : 

1.  Rub  the  face  of  a  coin  or  button  against  a  hard-wood 
board  for  a  minute  or  two.     The  metal  will  become  very 
hot. 

2.  Bend  a  piece  of  iron  wire  back  and  forth,  and  then 
feel  of  the  place  where   the  bending  occurred.     The  wire 
becomes  too  hot  to  hold  in  the  fingers. 

3.  Give   the  end  of   a  nail  or  a  piece  of  lead  a  dozen 
rapid  blows  witn  a  hammer.     A  decided  rise  in  temperature 
will  be  detected. 

4.  Place  some  tinder  in  the  end  of  the  piston  of  a  "fire 
syringe,"  Fig.  165,  and  then  force  the  piston  into  the  cylin- 
der.    When   the  piston  is  withdrawn,  the  tinder  will   be 

FIG.  165.   found  to  be  burning. 

These  cases  are  alike  in  that  the  heat  required  to  bring 
about  the  rise  in  temperature  is  produced  in  every  instance 
at  the  expense  of  work  on  the  part  of  the  person  perform- 
ing the  experiment.  Thus  energy  is  given  up  by  the  per- 
son, and  heat  appears.  Again,  a  moving  train  apparently 
loses  its  kinetic  energy  when  the  brakes  are  applied;  but 
an  examination  of  the  brakes  and  wheels  reveals  the  fact 
that  the  energy  has  been  converted  into  heat,  as  is  shown 
by  a  large  increase  in  temperature. 

On  the  other  hand,  heat  is  constantly  used  in  steam 
engines  for  hauling  trains,  running  machinery,  and  for 
performing  work  in  many  other  ways. 

The  conclusion  to  which  experimental  results  lead  is 
that  heat  is  a  form  of  energy.  The  steam  engine  is  simply 
a  device  for  converting  heat  energy  into  a  form  of  energy 
which  can  be  utilized,  i.e.  into  the  energy  of  mechanical 
motion.  When,  however,  this  motion  is  checked,  the 
mechanical  energy  is  changed  back  into  heat  energy. 


TEMPERATURE  CHANGES  AND  MEASUREMENT  211 

213.  Molecular  Theory  Applied  to  Heat.  —  The  explana- 
tion of  such  phenomena  as  we  have  before  us  is  greatly 
aided  by  the  so-called  molecular  theory  of  matter  which  was 
briefly  stated  in  §  129.     It  is  assumed  that  all  matter  is 
composed  of  small  particles  called  molecules.      The  molecules 
of  a  body  are  separated  by  small  spaces  within  which  they 
move  rapidly  about,  probably  with  frequent  collisions.     In 
solids  each  molecule  is  restricted  in  its  motion  to  a  certain 
space  which  it  does  not  leave.     When  this  restriction  is 
removed,  the  body  assumes  the  liquid  state  and  the  only 
constraint  is  the  mutual  attraction  between  the  molecules 
(cohesion).     In  gases  even  this  last  limitation  is  practi- 
cally removed,  and  hence  the  space  occupied  by  a  given 
mass  of  gas  is  governed  only  by  the  size  of  the  vessel 
containing  it. 

With  the  help  of  the  molecular  theory  it  is  now  pos- 
sible to  give  a  more  definite  idea  of  the  nature  of  heat. 
If  m  represents  the  mass  of  a  molecule  and  v  the  average 
velocity  of  the  molecules  of  a  body,  it  is  plain  that  each 
molecule  possesses  kinetic  energy  of  the  amount  ^mv2 
(Eq.  2,  §  62),  and  the  entire  body  will  have  within  it  as 
many  times  this  amount  as  there  are  molecules.  This 
energy  is  called  heat.  Hence,  we  may  define  heat  as  the 
kinetic  energy  of  molecular  motion. 

214.  Some  Effects  of  Temperature  Changes.  —  When  by 
increasing  or  decreasing  the  amount  of  heat  in  a  body  its 
temperature  is  raised  or  lowered,  one  or  more  resulting 
changes  may  take  place :    (1)  the  body  may  expand  or 
contract ;  (2)  the  body  may  undergo  a  change  in  its  prop- 
erties, as  in  hardness,  elasticity,  ductility,  etc.;    (3)  the 
body  may  change  its  pressure  against  other  bodies,   e.g. 
a   gas  may  increase  or  decrease  its  pressure  against  the 
walls  of  the  containing  vessel. 

The  application  of  heat  to  a  body,  however,  does  not 


212         A  HIGH   SCHOOL   COURSE   IN   PHYSICS 

always  change  its  temperature.  The  body  may  be  changed 
from  a  solid  to  a  liquid,  or  from  a  liquid  to  a  gas  while 
its  temperature  remains  constant,  e.g.  melting  ice,  boiling 
water,  etc.  These  different  classes  of  phe- 
nomena are  commonly  known  as  heat-effects. 
215.  Measurement  of  Temperatures.  — 
The  instrument  most  widely  used  for  the 
determination  of  temperatures  is  the  mer- 
curial thermometer.  It  is  constructed  upon 
the  principle  that  mercury  expands  when 
warmed.  This  thermometer  consists  of  a 
capillary  tube  at  the  lower  end  of  which  is 
a  bulb  containing  mercury,  Fig.  166.  The 
mercury  completely  fills  the  bulb  and  ex- 
tends some  distance  into  the  tube.  Since 
the  expansion  of  mercury  is  greater  than 
that  of  glass,  the  thread  of  mercury  in  the 
tube  rises  when  the  temperature  of  the 
mercury  in  the  instrument  is  increased,  and 
falls  when  it  is  decreased.  Before  the  tube 
of  a  thermometer  is  sealed,  the  mercury  in 
the  bulb  is  heated  until  it  entirely  fills  the 
instrument,  in  which  condition  the  glass 
at  A  is  sealed  off  in  a  hot  flame.  When 
the  mercury  cools  and  contracts,  it  leaves 
a  good  vacuum  above  it  in  the  tube. 
FIG.  166.  — Tube  216.  The  Fixed  Points  of  a  Thermometer. 
a^ier curial  — In  order  to  make  it  possible  to  compare 
Thermometer.  \\^Q  temperature  measurements  of  one  ther- 
mometer with  those  of  another,  two  fixed  points  that  are 
easily  obtained  are  located  on  the  tube  of  the  instrument. 
The  first  of  these  is  the  freezing  point  of  pure  water.  This 
point  is  found  by  packing  the  thermometer  in  ice  or 
snow,  as  shown  in  Fig.  167.  When  the  mercury  has 


TEMPERATURE  CHANGES  AND  MEASUREMENT  213 


ceased  to  fall,  the  position  of  the  end  of 
the  column  is  marked  upon  the  tube. 

The  second  fixed  temperature  point  is 
the  loiling  point  of  pure  water.  The  ther- 
mometer is  suspended  over  boiling  water 
in  a  tall  vessel  so  that  the  thread  of  mer- 
cury in  the  tube  is  completely  enveloped 
by  the  steam,  as  in  Fig.  168.  The  mer- 
cury rises  for  a  time,  but 
finally  comes  to  a  posi- 
tion at  which  it  remains 
stationary.  Since,  how- 
ever, this  temperature 
changes  with  the  pres- 
sure of  the  atmosphere, 
it  should  be  taken  under 
normal  atmospheric 
pressure  (i.e.  760  milli- 
meters of  mercury).  Otherwise  a  cor- 
rection must  necessarily  be  made. 

The  points  thus  obtained  on  a  ther- 
mometer scale   are   often  marked  with 
FIG.  i68.-Determin-    the  words   "freezing"    and  "boiling," 

ing    the    Boiling     ,,  ,     . 

Point  on  a  Tner-    these   being   the   names   given   to   the 

mometer  Tube.         nxe(j  points  of  temperature. 

217.    Graduation  of   Thermometer   Scales.  —  The    space 

between  the  freezing  and  boiling  points  is  now  divided 

into   temperature    units    called    degrees.      According   to 

the    centigrade1   scale    the   freezing   point  is  marked  0°, 

and   the  boiling  point  100°.     The  interval   between  the 

two  points  is  then  divided  into  100  equal  parts.     Similar 

divisions   are   produced   on  the   tube   above  the   boiling 

point  and   below  the   freezing  point.     Centigrade   ther< 

1  From  centum  and  gradus,  meaning  a  hundred  degrees. 


FIG.  167.  — Locat- 
ing the  Freez- 
ing Point  on  a 
Thermometer 
Tube. 


214 


A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


mometers  are  almost  exclusively  used  for  scientific  pur- 
poses. 

The  Fahrenheit  thermometer  scale  was  introduced  by  a 
German  physicist  of  that  name  about  1714.  On  this  scale 
the  freezing  point  is  marked  320,1  and  the 
boiling  point  212°.  The  interval  between 
„  these  two  points  is  therefore  divided  into  180 
equal  parts,  and  similar  divisions  are  laid 
off  both  above  the  boiling  point  and  below 
the  freezing  point.  The  Fahrenheit  ther- 
mometer is  the  household  instrument  in  use 
among  most  English-speaking  people,  and 
is  that  employed  by  the  United  States 
Weather  Bureau  and  by  physicians. 

On  all  thermometers,  temperatures  below 
the  zero  point  are  read  as  negative  quan- 
tities. In  every  case  the  initial  letter  of 
the  name  is  affixed  to  indicate  the  scale 
used.  For  example,  25°  C.,  100°  F. 

218.  Thermometer  Scales  Compared. — It 
is  obvious  that  any  thermometer  can  be  pro- 
vided with  both  the  centigrade  and  the 
Fahrenheit  scales,  as  shown  in  Fig.  169.  It 

FIG.  169.  — Centi- 
grade   and  will  readily  be  observed  that  100  centigrade 

Fahrenheit  denrrees  measure  the  same  interval  as  180 

Thermometer        ° 

Scales.  .Fahrenheit  degrees.     Hence, 


32C 


or 


100  centigrade  degrees  —  180  Fahrenheit  degrees, 
1  centigrade  degree    =  |  Fahrenheit  degree. 


But  in  order  to  change  a  temperature  reading  from  one 
system  into  the  other,  it  is  necessary  to  take  account  of 
the  fact  that  0°  F.  is  32  Fahrenheit  degrees  below  0°  C. 

1  Fahrenheit  placed  0  at  the  temperature  which  he  produced  by  a  mixture 
of  ice,  water,  and  salainnioniac. 


TEMPERATURE  CHANGES  AND  MEASUREMENT     215 

For  example,  68°  F.  is  68  -  32,  or  36  Fahrenheit  degrees 
above  the  freezing  point.  But  36  Fahrenheit  degrees  are 
equivalent  to  $  x  36,  or  20  centigrade  degrees.  Now  a 
temperature  that  is  20  centigrade  degrees  above  the  freez- 
ing point  is  20°  C.  Therefore  the  temperature  68°  F.  is 
equivalent  to  20°  C. 

Letting  F  represent  a  Fahrenheit  reading  and   C  the 
corresponding  reading  on  the  centigrade  scale,  we  have : 

F  -  32  =  f  C.  (l) 

219.  Range  of  a  Mercurial  Thermometer.  —  The  use  of  a 
mercurial  thermometer  in  the  measurement  of  high  and 
low  temperatures  is  limited  by  the  boiling  and  freezing 
points  of  mercury.     The  former  is  about  350°  C.,  and  the 
latter  —  38.8°  C.     The  boiling  of  the  mercury  can  be  pre- 
vented by  increasing  the  pressure  upon  it  by  the  presence 
of  nitrogen  gas  above  it  in  the  tube.     Such  thermometers 
may  register  temperatures  up  to  about  500°  C.    For  temper- 
atures below  —  39°  C.  liquids  having  low  freezing  points 
must  be  used.     Such  a  liquid  is  alcohol,  which  freezes  at 
—  111°  C.    Many  alcohol  thermometers  are  in  common  use. 

220.  Galileo's    Air    Thermometer.  —  The  ^ 
first    instrument    designed    for    the    meas- 
urement of  temperatures  was  Galileo's  air 
thermometer,  Fig.  170.     The  use  of  this  in- 
strument dates  from  1593.     The  device  con- 
sists of  a  vertical  glass  tube  of  small  bore, 

on  the  upper  end  of  which  is  a  large  bulb 
containing  air.  This  air  is  warmed  slightly, 
and  the  end  of  the  tube  is  placed  in  some 
liquid,  such  as  colored  water.  When  the 
air  cools,  the  liquid  rises  in  the  tube,  being 
forced  up  by  the  atmospheric  pressure.  Ob-  FlG- 170-  —  Gali- 

i      £       v        -j         i  -fi      •  j    c  n         leo'sAirTher- 

viously  the  liquid  column  will  rise  and  fall       mometer. 


216         A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

according  as  the  temperature  of  the  air  in  the  bulb  is  re- 
duced or  increased.  On  account  of  its  sensitiveness  to 
small  changes  of  temperature,  this  thermometer  is  fre- 
quently used  for  experimental  purposes. 

EXERCISES 

1.  Would  the  range  of  a  mercurial  thermometer  of  given  length 
be  increased  or  decreased  by  reducing  the  size  of  the  bulb  ?  by  mak- 
ing the  bore  of  the  tube  smaller?    Would  the  distance  representing 
a  degree  be  increased  or  decreased? 

2.  If  the  bulb  of  a  mercurial  thermometer  should  permanently 
contract  after  its  graduation,  how  would  the  fixed  points  be  affected  ? 

3.  In  a  certain  experiment  only  the  bulb  of  a  thermometer  is  ex- 
posed to  the  temperature  that  it  is  desired  to  measure.     If  this  tem- 
perature is  above  that  of  the  room,  will  the  reading  of  the  instrument 
be  too  large  or  too  small  ? 

4.  How  would  you  proceed  to  test  experimentally  the  points  on  a 
mercurial  thermometer  ? 

5.  The  boiling  point  of  water  falls  0.1  of  a  centigrade  degree  for 
a  decrease  of  0.27  cm.  in  the  atmospheric  pressure.     What  is  the 
boiling  point  when  the  barometer  reads  73.3  cm.  ?  A  ns.  99°  C. 

6.  Reduce  the  following  centigrade  readings  to  the  corresponding 
values  on  the  Fahrenheit  scale :  20°,  35°,  50°,  -20°,  -40°. 

7.  How  many  centigrade  degrees  lie  between  the  Fahrenheit  and 
centigrade  zero  marks? 

8.  The  following  temperature  measurements  were  taken  with  a 
Fahrenheit  thermometer:  77°,  41°,  14°,  -4°,  -40°.     What  would  a 
centigrade  thermometer  have  indicated  in  each  case  ? 

9.  The  difference  in  temperature  of  two  vessels  of  water  is  25 
centigrade  degrees.     Express  the  difference  in  Fahrenheit  units. 

10.  One   room   is   18.  Fahrenheit   degrees  warmer  than   another. 
What  is  the  difference  between  their  temperatures  on  the  centigrade 
scale  ? 

11.  The  boiling  point  of  water  at  a  certain  place  was  found  to  be 
98.8°  C.    What  was  the  atmospheric  pressure  at  the  time  ?    See  Exer.  5. 

Ans.  72.76  cm. 
2.    EXPANSION  OF  BODIES 

221.  Linear   Expansion.  —  We   have   already   observed 
the  use  made  of  the  expansion  of  mercury  when  heated. 


TEMPERATURE  CHANGES  AND  MEASUREMENT  217 


It  is  generally  true  of  all  bodies  that  an  increase  in  size 
accompanies  a  rise  in  temperature.  Thus  a  metal  rod,  for 
example,  undergoes  an  increase  in  length,  which,  although 
usually  small,  must  be  taken  into  account  in  planning 
bridges,  laying  railroad  tracks,  etc. 

1.  Figure  171  illustrates  the  well- 
known   ring   and   ball.      When    both 
are  of  the  same  temperature,  the  ball 
passes  readily  through  the  ring.  When, 
however,  the  ball  is  heated,  it  becomes 
too  large  to  fit  the  ring.     If  cooled,  it 
will  be  found  to  resume  its  original  size. 

2.  Let  a  metal  bar,  preferably  of  brass,  rest  at  one  end  upon 
a  block   of  wood,  as  A,  Fig.  172,  and  at  the   other  end  upon   a 
round  glass  or  metal  rod  B  about  2  millimeters  in  diameter  placed 


FIG.  171.  — Ring  and  Ball. 


FIG.  172.  —  Expansion  of  a  Metal  Rod. 

upon  a  smooth  block  of  wood.  Attach  a  very  light  index  of  glass  or 
paper  about  20  centimeters  long  to  the  small  rod  in  such  a  manner 
that  it  can  move  over  a  scale  C,  as  shown.  Now  heat  the  bar  by  mov- 
ing a  flame  along  it.  The  movement  of  the  index  will  indicate  an 
elongation  of  the  bar.  When  the  bar  cools,  the  index  returns  to  the 
original  position. 

222.  Coefficients  of  Linear  Expansion.  —  Not  all  solids 
expand  equally.  For  example,  a  bar  of  copper  a  meter 
long  expands  more  than  a  bar  of  iron  of  equal  length 
for  the  same  increase  in  temperature.  The  ratio  of  the 
increase  in  length  of  a  metal  bar  for  an  increase  of  one  degree 


218 


A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


in  temperature  to  its  length  at  0°  C.  is  called  the  coefficient 
of  linear  expansion  of  the  metal. 

Thus  a  rod  of  metal  one  meter  long  that  expands 
1  millimeter  when  the  temperature  rises  from  0°  C.  to 
100°  C.  has  a  coefficient  of  linear  expansion  equal  to  1  -r- 
(1000  x  100),  or  0.00001.  The  coefficient  of  linear  ex- 
pansion of  a  substance  is  expressed  by  the  equation 


where  ?a  and  Z2  are  the  lengths  before  and  after  heating, 
and  £j  and  t2  are  the  initial  and  final  temperatures. 

COEFFICIENTS  OF  LINEAR  EXPANSION 


Aluminium      .     .  0.000023 

Brass 0.000018 

Copper    ....  0.000017 

Glass 0.000009 

Invar  .  ,  0.00000087 


Iron 0.000012 

Lead 0.000027 

Platinum    ....  0.000009 

Steel  .  ,  0.000011 


223.  Applications  of  Unequal  Expansion.  —  By  referring  to 
the  table  of  the  coefficients  of  expansion  of  metals  given  in  §  222,  it 
will  be  seen  that  the  common  metals,  brass 
and  iron,  expand  unequally.  Hence  if  two 
flat  bars  of  these  metals  are  riveted  together, 
as  shown  in  (a),  Fig.  173,  an  increase  in  tern- 


the  iron  and  cause  the  bar  to  bend,  as  in 
The  bending  of  a  compound  bar  of  two  metals 
is  employed  in  the  dial  thermometer  in  common 
use.     The  bar  in  this  case  is  made  circular  in 
form  and  -has  one  end  fixed,  as  at  A,  Fig.  174. 
The  other  end  is  attached  by  means  of  a  cord 
or  chain  at  B  to  a  small  axle  (7,  which  carries 
the  pointer  D.     A  rise  in  temperature  causes 
the  free  end  of  the  bar  B  to  move  inward  and    FlG  174  _r)jai  Ther« 
the  pointer  to  register  a  higher  temperature  on  mometer. 


TEMPERATURE  CHANGES  AND  MEASUREMENT  219 

the  graduated  scale.     For  a  fall  in  temperature  the  reverse  movement 
takes  place. 

The  same  principle  is  ingeniously  applied  to  the  balance  wheel 
of  a  watch,  in  o^der  to  make  the  period 
of  vibration  independent  of  temperature 
changes.  An  increase  in  temperature  weak- 
ens the  hairspring  which  controls  the  vibra- 
tion of  the  wheel  and  lengthens  its  spokes, 
the  effect  in  each  instance  being  to  make 
the  watch  lose  time.  In  order  to  correct  this 
tendency,  the  balance  wheel  is  constructed 
as  shown  in  Fig.  175.  The  outer  rim  of  each 
of  the  compound  bars  A  and  B  is  made  of  FIG.  175.  —  Balance  Wheel 
brass,  and  the  inner  part  of  iron.  An  in-  of  a  Watch- 

crease  in  temperature  causes  the  portions  A  and  B  to  bend  toward 
the  center  just  enough  to  keep  the  period  of  vibration  constant. 

EXERCISES 

1.  Ascertain  how  steel  tires  are  tightened  or  "set "by  a  black- 
smith, and  explain  the  various  steps  of  the  process. 

2.  In  what  manner  do  engineers  take  account  of  the  expansion 
and  contraction  of  the  rails  when  laying  a  railroad  track  ? 

3.  Consult  the  table  of  linear  coefficients  of  expansion,  and  ascer- 
tain why  platinum  wires  can  be  sealed  in  glass  without  danger  of 
breakage  when  the  glass  cools.     Examine  an  incandescent  lamp  bulb, 
and  see  that  this  is  the  case. 

4.  Why  is  one  end  of  a  long  steel  bridge  often  supported  on  rollers  ? 

5.  Invar  is  an  alloy  of  nickel  and  steel.     Consult  the  table  of  linear 
coefficients  of  expansion,  and  show  why  it  is  a  valuable  metal  from 
which  to  make  tapes  for  measuring,  standards  of  length,  and  pendulum 
rods. 

6.  Glass  stoppers  can  often  be  loosened  by  carefully  heating  the 
neck  of  the  bottle.     Explain. 

7.  How  much  does  the  length  of  a  90-foot  steel  rail  vary  if  the 
extremes  of  temperature  are  -  24°  C.  and  35°  C.  ? 

8.  At  the  temperature  of  0°  C.  an  iron  pipe  is  100  ft.  long.     What 
will  be  its  length  when  steam  at  100°  C.  is  passing  through  it? 

9.  A  metal  rod  is  60  cm.  long  and  expands  1.02  mm.  when  the 
temperature  is  raised  from  0°  C.  to  100°  C.     Compute  the  coefficient 
of  linear  expansion  of  the  metal. 


220         A   HIGH   SCHOOL  COURSE   IN   PHYSICS 

224.  Cubical  Expansion.  —  In  general,  substances  when 
heated  expand  in  all  directions,  i.e.  a  rise  in  temperature 
is  accompanied  by  an  increase  in  volume.     This  may  be 
shown  to  hold  true  for  water   by  the  following   experi- 
ment: 

Fill  a  flask  with  water,  and  insert  a  rubber  stopper  through  which 
passes  a  small  glass  tube  about  40  centimeters  long.  Some  of  the 
liquid  will  rise  in  the  tube.  Mark  the  position  of  the  top  of  the 
liquid  column,  and  set  the  flask  in  a  vessel  of  warm  water.  The  liquid 
at  first  falls  slightly  in  the  tube  as  the  flask  expands  and  then  rises 
slowly  because  of  the  increase  in  volume  of  the  water. 

225.  Coefficients   of  Cubical   Expansion.  —  The   cubical 
expansion  of  a  substance  is  related  to  volumes  in  the  same 
manner  as  linear  expansion  is  related  to  lengths.     Thus 
the  coefficient  of  cubical  expansion  of  a  substance  is  the  ratio 
of  the  increase  in  volume  for  a  change  of  one  degree  in  tem- 
perature, to  the  volume  at  0°  O. 

The  cubical  expansion  of  a  substance  may  be  expressed 
by  the  equation 

k  =  Il^O,  (3) 

VQI 

where  vl  and  VQ  are  the  volumes  at  the  temperatures  t°  C. 
and  0°  C.  respectively.  This  equation  is  easily  reduced 
to  the  form 

Vl=v0(l+kt).  (4) 

The  coefficient  of  cubical  expansion  of  any  substance  is 
three  times  the  coefficient  of  the  linear  expansion  of  that 
substance.  Hence  the  cubical  expansions  for  the  sub- 
stances given  in  the  table  in  §  222  are  easily  computed. 

226.  Abnormal  Expansion  of  Water.  —  If  a  quantity  of 
water   at   freezing   temperature    is    warmed,    its   volume 
decreases  until  it  reaches  a  temperature  of  4°  C.     Upon 
being   heated   above   this   point  it  expands  as   do  other 


TEMPERATURE  CHANGES  AND  MEASUREMENT  221 

liquids.  This  exception  to  the  general  rule  is  of  vast 
importance  in  nature.  As  the  atmosphere  grows  colder, 
the  water  at  the  surface  of  lakes  and  ponds  falls  in  tem- 
perature, and  its  density  at  first  increases.  The  surface 
layers  of  water  then  sink  and  are  replaced  by  the  warmer 
water  from  below.  This  continues  until  the  temperature 
of  4°  C.  is  reached,  when  further  cooling  makes  the  sur- 
face water  less  dense  than  that  underneath.  Hence  the 
water  that  has  cooled  below  4°  C.  does  not  sink,  but 
remains  at  the  top  and  is  frozen.  Thus,  since  practically 
the  entire  quantity  of  water  in  a  lake  must  be  reduced  to 
4°  C.  before  the  surface  is  frozen,  ice  is  much  slower  in 
forming  on  deep  bodies  of  water  than  on  shallow  ones. 
Furthermore,  on  account  of  the  fact  that  the  colder  ice 
and  water  at  the  top  do  not  transmit  the  heat  rapidly 
away  from  the  warmer  layers  of  the  liquid  below,  the 
unfrozen  water  remains  at  a  temperature  of  about  4°  C. 
Hence,  animal  life  flourishes  at  the  bottom  of  a  lake  in 
winter,  even  when  a  little  lower  temperature  would  prove 
fatal. 

227.  Cubical  Expansion  of  Gases.  —  When  the  temper- 
ature of  a  gas  is  raised,  its  volume  is  increased,  unless 
the  expansion  is  prevented  by  an  increase  of  the  pres- 
sure upon  the  gas.  Under  a  constant  pressure  the  ex- 
pansion of  gases  is  much  greater  than  that  of  liquids  or 
solids.  When  the  temperature  of  a  gas  is  increased,  the 
change  is  simply  an  increase  in  the  average  speed  of  its 
molecules.  As  a  consequence,  there  results  an  increase  in 
the  force  delivered  by  each  molecule  in  its  collision  with 
the  sides  of  the  containing  vessel.  There  will  also  be  an 
increase  in  the  number  of  blows  delivered  per  second. 
Now  the  pressure  of  the  gas  outward  is  nothing  more 
than  the  result  of  this  continuous  bombardment  of  mole- 
cules ;  and  thus  an  increase  in  the  average  molecular 


222         A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

speed  produces  a  corresponding  increase  in  the  pressure 
of  the  gas.  By  permitting  the  gas  to  expand,  the  pres- 
sure may  be  kept  constant.  Under  a  constant  pressure, 
all  gases  have  the  same  coefficient  of  cubical  expansion  (Law 
of  Charles1).  This  number  has  been  found  experimen- 
tally to  be  2^,  or  0.00366,  of  the  volume  of  the  gas  at 
0°  C.  Thus  a  gas  whose  volume  at  0°  C.  is  100  cubic  cen- 
timeters, when  heated  to  25°  C.  increases  in  volume  -ff^ 
of  100  cubic  centimeters.  Its  volume  becomes,  therefore, 
100  +  ffig  x  100,  or  109.1  cubic  centimeters. 

228.  The  Absolute  Scale  of  Temperatures.  —  If  a  body  of 
air,  for  example,  at  0°  C.  is  kept  at  a  constant  pressure 
and  heated,  its  volume  will  increase  -%fa  of  its  original 
volume  for  every  degree  that  its  temperature  is  raised. 
Thus,  at  273°  C.  its  volume  will  be  doubled.  On  the  other 
hand,  if  the  original  body  of  air  is  cooled  below  0°,  its 
volume  will  be  diminished  ^3-  of  its  volume  at  0°  for  every 
degree  that  its  temperature  is  lowered.  If,  now,  the  volume 
were  to  continue  to  diminish  at  this  rate  until  the  tem- 
perature should  reach  —  273°  C.,  mathematically  it  would 
become  nothing.  Practically,  however,  the  air  and  other 
gases  become  liquids  before  reaching  this  temperature,  and 
thus  lose  the  properties  of  gases.  A  temperature  273  cen- 
tigrade degrees  below  the  freezing  point  of  water  is  called 
absolute  zero,  and  temperatures  measured  from  this  point 
as  the  0  of  the  scale  are  called  absolute  temperatures.  It  is 
clear  that  temperatures  on  the  centigrade  scale  can  be  re- 
duced to  the  absolute  by  simply  adding  273°.  Thus  20°  C. 
=  293°  Ab.,  and  -  40°  C.  =  233°  Ab. 

Although  no  one  has  ever  succeeded  in  cooling  a  body 
to  absolute  zero,  temperatures  approaching  within  a  very 
few  degrees  of  this  point  have  been  attained  by  the  evapo- 
ration of  liquefied  gases.  The  following  table  will  serve 
1  Also  called  Gay-Lussac's  Law. 


TEMPERATURE  CHANGES  AND  MEASUREMENT  223 


to  show  some  facts  regarding  the  history  of  low  tempera- 
ture production : 


DATE 

TEMPERATURE 

EXPERIMENTER 

1714 

_  170  c  

Fahrenheit 

1778 

_  40°   C  

Van  Marum 

1823 

102°  C  

Faraday 

1877 

_  103°  C  

Cailletet 

1877 

_183°C  

Pictet 

1898 
1908 

-262°C  
269°  C  

Dewar 
Onnes 

229.  Laws  of  Gaseous  Bodies.  —  Since  the  volume  of  a 
gas  is  doubled  when  its  temperature  is  raised  from  273° 
Ab.  (0°C.)  to  2  x  273,  or  546°  Ab.  (273°  C.),  and  the 
increase  in  volume  is  uniform  (§  227),  it  is  clear  that  the 
following  law  may  be  stated  : 

The  volume  of  a  given  mass  of  gas  under  constant  pressure 
is  proportional  to  its  absolute  temperature. 

Again,  if  the  body  of  gas  is  confined  in  a  vessel  of  suffi- 
cient rigidity  to  keep  the  volume  constant  at  all  tempera- 
tures, the  pressure  of  the  gas  against  the  walls  of  the 
vessel  will  increase  as  the  temperature  rises.  Further- 
more, the  pressure  will  increase  gr^-  of  the  pressure  at 
0°  C.  for  every  degree  that  the  temperature  is  raised,  and 
will  decrease  g-fj  of  the  pressure  at  0°C.  for  every 
degree  that  the  temperature  is  lowered.  In  other  words, 
when  the  volume  of  a  gas  remains  constant,  the  change  in 
pressure  takes  place  according  to  a  law  similar  to  that 
governing  the  change  in  volume.  Hence 

The  pressure  of  a  given  mass  of  gas  whose  volume  remains 
constant  is  proportional  to  the  absolute  temperature. 


224         A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

EXAMPLE.  —  At  25°  C.  the  volume  of  a  certain  mass  of  gas  is  400 
cm.8.  Compute  its  volume  when  the  temperature  is  lowered  to  0°  C. 
and  the  pressure  kept  constant.  If  the  original  pressure  is  740  mm., 
compute  the  pressure  after  the  decrease  in  temperature,  assuming 
that  the  volume  remains  constant. 

SOLUTION.  —  Letting  x  be  the  volume  of  the  gas  at  0°  C.,  we  have, 
by  applying  the  law  of  volumes  stated  above, 

400  :  x  :  :  25  +  273  :  273 ; 
whence,  x  =  366.4  cm.8. 

In  the  second  place,  by  applying  the  law  of  pressures,  if  x  is  the 
pressure  at  0°  C., 

740  :  x  :  :  25  +  273  :  273 ; 

whence,  x  =  677.9  mm. 

230.  Laws  of  Charles  and  Boyle  Combined.  —  Boyle's 
Law  (§  149)  states  that  the  product  of  the  pressure  and 
volume  of  a  given  mass  of  gas  remains  constant  when 
the  temperature  remains  the  same.  In  practice,  volume, 
pressure,  and  temperature  may  all  vary.  In  such  cases 
the  following  law  will  hold : 

The  product  of  the  pressure  and  volume  of  a  given  mass  of 
gas  is  proportional  to  the  absolute  temperature. 

EXAMPLE.  —  The  volume  of  a  gas  collected  in  a  vessel  under  a 
pressure  of  740  mm.  and  at  a  temperature  of  20°  C.  is  500  cm.8.  Com- 
pute the  volume  that  the  gas  would  have  at  0°  C.  and  under  a  pressure 
of  760  mm. 

SOLUTION.  —  Let  x  be  the  required  volume.  By  combining  the 
laws  of  Boyle  and  Charles  we  have 

740  x  500  :  760  x  x ::  20  +  273  :  273 ; 
whence,  x  =  453.6  cm.8. 

EXERCISES 

1.  What  fractional  part  of  its  volume  at  0°  C.  does  a  cubic  meter 
of  gas  expand  when  warmed  from  that  temperature  to  50°  C.,  the 
pressure  remaining  constant?  What  is  the  final  volume  of  the  gas? 


TEMPERATURE  CHANGES  AND  MEASUREMENT  225 

2.  The  volume  of  a  certain  gas  at  20°  C.  is  300  cm.8.     What  is  its 
volume  when  the  temperature  is  reduced  to  0°  C.,  the  pressure  being 
constant  ? 

3.  The  pressure  exerted  by  a  gas  confined  in  a  reservoir  is  500  g. 
per  square  centimeter  when  the  temperature  is  10°  C.     What  is  its 
pressure  when  the  temperature  is  raised  to  40°  C.  ? 

4.  The  volume  of  a  gas  collected  in  a  chemical  experiment  is 
30  cm.8,   its  temperature  25°,  and  its  pressure  750  mm.     Find  the  vol- 
ume of  the  same  gas  at  0°  C.  and  under  a  pressure  of  760  mm. 

5.  If  the  mass  of  a  cubic  centimeter  of  air  at  0°  C.  and  under  a 
pressure  of  760  mm.  is  0.001293,  what  will  be  its  density  in  a  room 
where  the  temperature  is  22°  C.  and  the  barometer  reads  745  mm.  ? 

6.  The  quantity  of  air  in  a  room  8  x  12  x  15  ft.  will  contract  to 
what  volume  when  its  temperature  falls  from  20°  C.  to  0°  C.  ? 

7.  To  what  temperature  would  the  air  confined  in  a  flask  at  atmos- 
pheric pressure  and  a  temperature  of  10°  C.  have  to  be  heated  in  order 
to  exert  a  pressure  of  3.5  atmospheres? 

8.  Upon  heating  300  cm.8  of  a  gas  from  0°  C.  to  30°  C.,  the  volume 
was  found  to  be  333  cm.8.     Ascertain  the  coefficient  of  expansion  of 
the  gas. 

9.  The  capacity  of  a  steel  gas  cylinder  is  3  cu.  ft.     Illuminating 
gas  is  compressed  in  the  cylinder  until  the  pressure  is  15  atmospheres 
at  a  temperature  of  10°  C.     What  volume  will  this  quantity  of  gas 
assume  when  allowed  to  escape  into  a  space  where  the  pressure  is 
1  atmosphere  and  the  temperature  25°  C.  ? 

3.    CALORIMETRY,    OR   THE    MEASUREMENT   OF  HEAT 

231.  The  Unit  of  Heat. —The  unit  employed  in  the 
measurement  of  heat  is  called  the  calorie.  The  calorie  is 
the  quantity  of  heat  required  to  raise  the  temperature  of  a 
gram  of  water  one  centigrade  degree.  When  the  tempera- 
ture of  a  gram  of  water  is  raised  from  0°  C.  to  100°  C., 
100  calories  of  heat  are  required.  Again,  to  change  the 
temperature  of  1  kilogram  of  water  from  20°  C.  to  45°  C. 
requires  that  1000  x  (45  -  20),  or  25,000  calories  of  heat 
be  taken  on  by  the  water.  Thus,  if  a  mass  of  water  be 
changed  in  temperature  a  given  amount,  the  quantity  of 

heat  involved  is  measured  ~by  the  product  of  the  mass  of  water 
1ft 


226         A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

and  its  change  of  temperature.  Hence,  to  change  the 
temperature  of  a  mass  of  m  grams  of  water  t  centigrade 
degrees,  we  may  write  for  the  required  quantity  of  heat, 

H  (in  calories)  = 

m  (in  grams)  x  t  (in  centigrade  degrees).   (5) 

232.  Specific  Heat.  — If  a  given  quantity  of  heat  be  ap- 
plied to  equal  masses  of  different  kinds, of  matter,  as  lead, 
mercury,  water,  iron,  and  copper,  the  temperatures  will  not 
all  be  changed  equally.  The  amount  of  heat  that  will 
warm  a  gram  of  water  one  degree  (i.e.  one  calorie)  will 
raise  the  temperature  of  an  equal  mass  of  lead  or  of 
mercury  about  30  degrees  and  that  of  the  iron  or  the 
copper  about  10  degrees. 

Place  100  grams  each  of  lead  shot,  iron  cuttings,  and  bits  of  alumin- 
ium wire  in  three  large  test-tubes.  Set  the  tubes  upright  in  a  vessel 
of  boiling  water,  and  allow  them  to  remain  there  several  minutes 
while  the  .water  continues  to  boil.  Also  place  100  grams  of  water  at 
the  temperature  of  the  room  in  each  of  three  beakers.  Now  pour  the 
lead  shot  whose  temperature  is  100°  C.  into  the  water  in  one  of  the 
beakers,  stir  the  mixture  thoroughly,  and  ascertain  the  rise  in  temper- 
ature of  the  water.  Do  the  same  with  the  other  metals.  Although 
the  metals  fall  in  temperature  almost  equal  amounts,  they  deliver  to 
the  water  very  unequal  quantities  of  heat.  The  aluminium  will  warm 
the  water  through  about  twice  as  many  degrees  as  the  iron  and  about 
six  times  as  many  as  the  lead. 

Experiments  like  those  just  described  lead  to  the  con- 
clusion that  each  gram  of  lead  gives  out,  upon  cooling  one 
degree,  about  one  sixth  as  much  heat  as  one  gram  of 
aluminium.  Also  that  a  gram  of  iron  delivers  only  one 
half  as  much  heat  as  an  equal  mass  of  aluminium  when 
cooled  an  equal  amount.  For  this  reason  substances  are 
said  to  differ  in  thermal  capacity,  or  specific  heat. 

The  specific  heat  of  a  substance  is  the  ratio  of  the  quantity 
of  heat  required  to  raise  the  temperature  of  a  certain  mass 


TEMPERATURE  CHANGES  AND  MEASUREMENT  227 

of  it  one  degree  to  the  quantity  of  heat  required  to  raise  the 
temperature  of  an  equal  mass  of  water  one  degree. 

The  specific  heat  of  a  substance  is  numerically  equal  to 
the  number  of  calories  received  by  one  gram  of  the  sub- 
stance when  its  temperature  rises  1°  C.  Thus  the  heat 
required  to  raise  the  temperature  of  1  g.  of  iron  1°  C.  is 
0.113  calories;  of  1  g.  of  lead,  0.032  calories,  etc. 

233.  Specific  Heat  Determined.  —  When  a  hot  substance, 
as  heated  mercury,  for  example,  is  placed  in  cold  water, 
the  two  bodies  assume  the  same  temperature.  The  heat 
given  up  by  the  substance  which  cools  is  utilized  in  raising 
the  temperature  of  the  water.  In  other  words,  the  quantity 
of  heat  gained  by  the  cold  body  equals  that  lost  by  the  warm 
body. 

Place  300  grams  of  lead  shot  in  a  large  test-tube,  and  suspend  the 
tube  in  boiling  water  for  at  least  ten  minutes.  While  the  lead  is 
heating,  the  tube  should  be  kept  closed  with  a  cork.  Place  100  grams 
of  water  in  a  beaker,  and  cool  it  a  few  degrees  below  the  temperature 
of  the  room.  Now  pour  the  shot  quickly  into  the  water  and  stir  care- 
fully until  the  temperature  of  the  mixture  becomes  stationary.  Note 
and  record  the  rise  in  temperature  of  the  water.  In  order  to  complete 
the  solution  of  the  problem,  we  must  form  an  equation  that  expresses 
the  equality  between  the  heat  lost  by  the  lead  and  that  gained  by  the 
water.  The  following  illustration  will  make  the  process  clear: 

In  an  experiment  the  initial  temperature  of  the  water  was  17°  C., 
the  temperature  of  the  mixture  24.2°  C.,  and  the  boiling  point  100°  C. 

Let  x  be  the  specific  heat  of  lead. 

The  fall  in  temperature  of  the  lead     =  100°  -  24.2°. 

The  heat  lost  by  the  lead  =  (100  -  24.2)  300  a;  calories. 

The  rise  in  temperature  of  the  water  =  24.2°  -  17°. 

The  heat  gained  by  the  water  =  (24.2  -  17)  100  calories. 

Hence,  (100  -  24.2)  300  x  =  (24.2  -  17)  100 ; 

whence,  x  -  0.0316. 

This  process  for  determining  the  specific  heat  of  a  sub- 
stance is  known  as  the  "method  of  mixtures."  It  can  be 


228         A  HIGH   SCHOOL   COURSE   IN   PHYSICS 

applied  successfully  to  a  large  number  of  substances  that 
do  not  dissolve  when  placed  in  water.  The  specific  heats 
of  some  common  substances  are  given  in  the  following 
table: 

SPECIFIC  HEATS 

Aluminium      ....  0.212  Mercury 0.033 

Copper 0.093  Platinum 0.032 

Ice 0.502  Silver .  0.050 

Iron 0.113  Tin 0.055 

Lead 0.032  Zinc 0.093 

EXERCISES 

1.  A  mass  of  75  g.  of  water  is  cooled  from  95°  C.  to  32°  C.     How 
much  heat  is  given  up  ? 

2.  A  100-gram  mass  of  copper  rises  in  temperature  from  15°  C.  to 
100°  C.     How  much  heat  does  it  absorb? 

3.  If  500  calories  are  applied  to  500  g.  of  mercury  at  10°  C.,  to 
what  point  will  the  temperature  of  the  mercury  rise? 

SUGGESTION.  —  Let  x  be  the  final  temperature  of  the  mercury. 

4.  A  mass  of  iron  weighing  400  g.  and  having  a  temperature  of 
98°  C.  is  placed  in  100  g.  of  water  at  14°  C. ;   the  temperature  of  the 
combined  masses  is  40°  C.     Compute  the  specific  heat  of  the  metal. 

SUGGESTION.  —  See  example  given  in  §  233. 

5.  Find  the  resulting  temperature  when  400  g.  of  water  at  90°  C. 
are  mixed  with  150  g.  of  water  at  10°  C. 

SUGGESTION.  —  If  a:  is  the  resulting  temperature,  the  former  mass 
falls  in  temperature  90  —  x  degrees,  while  the  latter  rises  x  —  10 
degrees.  Express  the  equality  between  the  heat  lost  by  the  former 
and  that  gained  by  the  latter. 

6.  A  vessel  contains  250  g.  of  lead  shot  and  150  g.  of  water.     Find 
the  amount  of  heat  required  to  raise  the  temperature  of  the  mixture 
from  15°  C.  to  80°  C.  An*.  10,270  calories. 

7.  The  temperature  of  a  block  of  ice  weighing  100  kg.  rises  from 
—  15°  C.  to  the  melting  point.     What  quantity  of  heat  is  absorbed  ? 

8.  If  100  g.  of  aluminium  at  97°  C.  were  dropped  into  50  g.  of 
water  at  10°  C.,  what  would  be  the  temperature  of  the  mixture? 


TEMPERATURE   CHANGES   AND  MEASUREMENT     229 

SUMMARY 

1.  Temperature  is  the  degree  of  hotness  of  a  body  and 
is   the  condition  which   determines  in  what   direction  a 
transfer  of  heat  will  take  place  (§§  210  and  211). 

2.  Heat  is  a  form  of  energy  into  which  all  other  forms 
are  convertible.     It  is  the  kinetic  energy  of  the  moving 
molecules  of  a  body  (§§  212  and  213). 

3.  Changes   in    the    temperature    of    bodies    produce 
(1)  expansion  and  contraction,  (2)  change  in  their  prop- 
erties, and  (3)  changes  in  pressure.     The  application  of 
heat  may  change  the  state  of  matter  without  producing  a 
change  in  temperature  (§  214). 

4.  The  mercury  thermometer  makes  use  of  the  expan- 
sion of   mercury  in   the   measurement   of   temperatures. 
Each  instrument  is  graduated  according  to  the  position  of 
two  fixed  points,  the  boiling  point  and  the  freezing  point 
of  pure  water  (§§  215,and  216). 

5.  On  the  centigrade  scale  the  freezing  point  is  marked 
0°,  and  the  boiling  point  (under  a  pressure  of  760  mm.) 
is  marked  100°.     On  the  Fahrenheit  scale  the  correspond- 
ing points  are  marked  32°  and  212°.     The  relation  be- 
tween temperatures  expressed  on  the  two  scales  is 

F-  32  =  f  O  (§§  217  and  218). 

6.  The  coefficient  of  linear  expansion  of  a  body  is  the 
ratio  of  its  increase  in  length  for  an  increase  of  1°  C.  to 
its  length  at  0°  C.  (§§  221  to  223). 

7.  The  coefficient  of  cubical  expansion  of  a  substance  is 
the  ratio  of  its  increase  in  volume  for  a  change  of  1°  C. 
to  its  volume  at  0°  C.  (§  225). 

8.  The  volume  of  a  given  mass  of  water  when  heated 
from  0°  C.  contracts  until  the  temperature  reaches  4°  C. 
Further  heating  causes  it  to  expand.     Hence  the  greatest 
density  of  water  is  at  4°  C.  (§  226). 


230         A  HIGH   SCHOOL  COURSE   IN    PHYSICS 

9.  The  coefficient  of  cubical  expansion  of  all  gases  is 
practically  the  same  and  is  expressed  by  the  fraction  ^j-g 
(§227).  " 

10.  Absolute  temperatures  are  measured  from  absolute 
zero,  which  is  the  same  as  —  273°  C.     Hence  temperatures 
measured  on  the  centigrade  scale  are  reduced  to  the  abso- 
lute by  adding  273°  (§  228). 

11.  The  volume  of  a  given  mass  of  gas  under  constant 
pressure    is    proportional    to    its    absolute    temperature 
(§  229). 

12.  The  pressure  of   a   gas  under  constant  volume  is 
proportional  to  its  absolute  temperature  (§  229). 

13.  The  product  of  the  pressure  and  volume  of  a  given 
mass  of  gas  is  proportional  to  its  absolute  temperature 
(§  230). 

14.  Heat   is  expressed   in  terms  of   a  unit  called   the 
calorie.     The  calorie  is  the  quantity  of  heat  required  to 
raise  the  temperature  of  1  g.  of  water  1°  C.  (§  231). 

15.  Like  masses  of  different  substances  require  different 
quantities  of   heat  to  produce  equal  changes  of  temper- 
ature, i.e.  they  differ  in  specific  heat.    The  specific  heat  of 
a  substance  is  the  ratio  of  the  quantity  of  heat  required 
to  raise  the  temperature  of  a  certain  mass  of  it  1°  C.  to 
the   quantity  required    to  raise   the    temperature  of   an 
equal  mass  of  water  the  same  amount  (§  232). 

16.  The  quantity  of   heat  required   to  raise  the  tem- 
perature of  any  mass  of  a  substance  a  given  amount  is  the 
product   of   the  mass,  the   rise  in   temperature,  and  the 
specific  heat  of  the  substance  (§  233). 

17.  Specific   heat   is   determined   by   the    "method  of 
mixtures"  (§233). 


CHAPTER   XII 

HEAT:    TRANSFERENCE   AND   TRANSFORMATION   OF 
HEAT   ENERGY 

1.    CHANGE  OF  THE   MOLECULAR  STATE   OF  MATTER 

234.  Fusion.  —  If  the  motion  of  the  molecules  of  a  body 
is  principally  of  a  vibratory  nature,  we  find  difficulty  in 
changing  the  form  of  the  body  and,  therefore,  call  it  a 
solid.  When,  however,  by  the  application  of  heat,  the 
molecular  motion  is  so  increased  that  the  particles  break 
away  from  the  constraining  forces  and  thus  overcome 
their  "fixedness  of  position,"  the  mass  assumes  the  prop- 
erty of  fluidity  and  becomes  a  liquid.  Hence  the  state  in 
which  a  substance  exists  depends  largely  upon  its  tem- 
perature. Thus  mercury,  which  is  a  liquid  under  ordinary 
conditions,  changes  into  a  gas  at  350°  C.  and  remains  a 
solid  at  all  temperatures  below  —  39°  C.  The  temper- 
ature at  which  a  solid  changes  into  a  liquid  is  called  its 
melting  or  fusing  point,  and  the  process  is  called  fusion. 
The  reverse  change,  i.e.  from  a  liquid  to  a  solid,  is  called 
solidification.  In  the  case  of  water,  the  process  is  called 
freezing. 

Collect  a  quantity  of  snow  in  a  vessel,  preferably  when  the  out-door 
temperature  is  several  degrees  below  the  freezing  point.  In  summer 
broken  ice  must  be  usedv  Place  a  thermometer  in  the  snow  so  that  it 
can  be  read  easily  and  set  the  vessel  outside  the  window.  When  ready 
to  proceed  with  the  experiment,  bring  the  snow  into  the  warm  room, 
and  read  the  thermometer  at  brief  intervals.  The  temperature  of  the 
snow  will  be  found  to  rise  gradually  to  0°  C.,  where  the  mercury  re- 
mains while  the  snow  is  melting.  The  vessel  of  snow  may  be  gently 
heated,  and,  if  the  mixture  is  kept  thoroughly  stirred,  the  temperature 

231 


232          A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

will  remain  0°  C.     Continued  heating  after  the  snow  has  melted  will 
raise  the  temperature  of  the  resulting  liquid. 

The  melting  point  of  a  crystalline  substance,  as  snow,  is 
well  marked,  but  amorphous  bodies,  as  glass,  tar,  pitch, 
etc.,  pass  through  a  semiliquid  state  several  degrees  below 
the  temperature  of  liquefaction.  It  is  due. to  this  property 
that  glass  can  be  formed  into  vessels  of  any  desired  shape, 
and  that  wrought  iron  can  be  fashioned  according  to  the 
demands  of  the  blacksmith. 

235.  Laws  of  Fusion.  —  The  laws  governing  the  fusion 
of  crystalline  substances  and  the  reverse  change  of  solidifi- 
cation are  as  follows : 

(1)  A  crystalline  substance  under  a  constant  pressure  has 
a  definite  fusing  point  which  is  also  the  temperature  at  which 
solidification  takes  place. 

(2)  When  a  crystalline  substance  begins  to  melt,  its  tem- 
perature remains  constant  until  all  of  it  is  liquefied. 

(3)  A  substance  that  contracts  while  melting  has  its  fusing 
point  slightly  lowered  by  increased  pressure,  but  a  substance 
that  expands  while  melting  has  its  fusing  point  slightly  raised 
by  an  increase  of  pressure. 

(4)  In   the  presence  of  a  dissolved   substance,  the  solid 
forms  by  crystallization  at.  a  temperature  below  the  freezing 
point  of  the  pure  solvent. 

MELTING  POINTS 

Mercury    ...     -  39.°  C:        Tin 232°  C. 

Ice 0.°  C.        Lead 325°  C. 

Benzine     ....      7.°  C.        Silver 954°  C. 

Acetic  acid    .     .     .     16.°  C.        Copper 1100°  C. 

Paraffin     ....     55.°  C.  Cast  iron   ....  1200°  C. 

236.  Effect  of  Pressure  on  the  Fusing  Point  of  Ice.  —  It 

is  well  known  that  water  expands  when  it  freezes.     This 


HEAT:  TRANSFERENCE  AND  TRANSFORMATION   233 

is  shown  by  the  floating  of  ice  and  the  bursting  of  frozen 
water  pipes.  Ice  therefore  contracts  when  it  melts. 
Hence,  according  to  the  third  law  of  fusion  stated  in 
§  235,  an  increase  of  pressure  lowers  the  melting  or  fusing 
point.  The  lowering  is  very  slight,  amounting  to  about 
0.0075°  C.  for  an  increase  of  one  atmosphere. 

1.  Let  two  pieces  of  ice  be  pressed  firmly 
together.     When  the  pressure  is  removed,  the 
pieces  will  be  found  to.  be  frozen  together. 

2.  Connect  two  heavy  weights  by  means  of 
a  strong  wire,  and  hang  them  over  a  block  of 
ice,  as  shown  in  Fig.  176.     In  a  short  time  the 
wire  will  cut  into  the  block  and,  at  last,  en- 
tirely through,  leaving  the  ice  still  in  one  piece.  FlG-  176-  ~  Wire  Cutting 

through  a  Block   of 

In  each  of  these  cases  the  melting  ice. 
point  at  the  places  where  the  pressure  is  applied  is  slightly 
lowered ;  hence  some  of  the  ice  melts,  forming  a  film  of 
water  a  little  below  0°  C.  When  the  pressure  is  removed 
the  film,  which  is  at  a  temperature  below  the  freezing  point, 
solidifies,  cementing  the  two  pieces  of  ice  together.  In  Ex- 
periment 2  the  melting  under  pressure  takes  place  below 
the  wire ;  then,  as  the  liquid  flows  up  above  the  wire,  the 
pressure  is  removed,  and  it  freezes.  This  process  is 
known  as  regelation  (pronounced  re  ge  Id'tiori). 

237.  Heat  of  Fusion.  —  In  general,  the  fusion  of  any 
solid  requires  the  application  of  heat.  If  the  substance 
is  of  a  crystalline  structure,  as  ice,  the  heat  energy  im- 
parted to  it  does  not  sensibly  raise  its  temperature  during 
the  melting  process.  In  all  such  cases  the  energy  sup- 
plied to  the  solid  is  used  in  producing  the  change  of  state. 
Non-crystalline  solids,  such  as  Waxes,  iron,  glass,  etc., 
become  plastic  when  heated,  and  have  no  definite  melting 
point. 

1.  Note  the  temperature  of  snow  or  finely  chipped  ice  placed  in  a 
metal  vessel.  Place  a  flame  under  the  vessel,  and  allow  some  of  the 


234          A  HIGH   SCHOOL  COURSE   IN    PHYSICS 

ice  to  melt.  Remove  the  flame,  stir  the  contents  of  the  vessel  well, 
and  again  take  the  temperature.  Apply  more  heat,  and  note  the 
temperature  after  stirring.  In  every  case  the  temperature  will  be 
found  to  be  the  same. 

2.  Place  equal  quantities  of  ice  and  ice  water  in  two  similar  vessels, 
and  set  both  in  a  large  vessel  of  hot  water  placed  over  a  flame.  In  one 
vessel  heat  is  changing  ice  into  water;  in  the  other  the  temperature 
of  the  water  is  being  raised.  If  the  contents  of  the  vessels  are  con- 
tinually stirred,  it  will  be  found  that  when  the  ice  is  melted,  the  tem- 
perature of  the  water  will  have  risen  to  about  80°  C. 

Heat  energy  is  applied  about  equally  to  the  ice  and  ice 
water  in  Experiment  2.  That  applied  to  the  cold  water 
increases  the  average  kinetic  energy  of  the  molecules, 
(§  213),  and  thus  the  temperature  is  raised.  The  heat 
applied  to  the  ice,  however,  does  not  produce  any  increase 
in  the  kinetic  energy,  but  suffers  a  transformation  into 
potential  energy  by  producing  molecular  separation  in  op- 
position to  the  mutual  attraction  (cohesion)  between  the 
particles  that  compose  the  body.  In  other  words,  the  heat 
energy  expended  in  melting  the  ice  has  ceased  to  be  heat  and 
simply  represents  the  work  necessary  to  change  the  ice 
from  the  solid  to  the  liquid  state.  The  number  of  calories 
per  gram  required  to  liquefy  a  substance  without  producing 
any  change  in  its  temperature  is  called  the  heat  of  fusion1 
of  that  substance. 

238.  Heat  of  Fusion  of  Ice  Measured.  —  The  quantity  of 
heat  required  to  melt  a  gram  of  ice  can  be  measured  by 
the  method  of  mixtures  as  shown  in  the  following  experi- 
ment: 

Let  300  grams  of  water  at  about  35°  C.  be  placed  in  a  beaker  and 
its  temperature  accurately  noted.  Prepare  also  a  quantity  of  ice  in 
lumps  about  as  large  as  walnuts.  Dry  the  pieces  of  ice  with  a  towel, 
and  drop  them  in  small  quantities  into  the  warm  water.  Stir 
thoroughly  to  melt  the  ice.  When  enough  ice  has  been  added  and 

1  Sometimes  called  "  latent "  heat  of  fusion. 


HEAT:  TRANSFERENCE   AND  TRANSFORMATION       235 

melted  to  make  the  temperature  of  the  water  about  5°C.,  weigh  the 
contents  of  the  beaker  and  compute  the  mass  of  ice  melted.  An  equa- 
tion is  now  formed  between  the  heat  lost  by  the  water  and  that  gained 
by  the  ice.  An  illustration  will  make  the  process  clear. 

In  an  experiment  the  temperature  of  the  warm  water  was  36.5°  C. 
After  adding  110  grams  of  ice  the  resulting  temperature  of  the  water 
was  5.5°  C. 

Let  x  be  the  heat  of  fusion  of  ice. 

The  heat  lost  by  the  warm  water  =  300  (36.5  -  5.5)  calories. 

The  heat  required  to  warm  110 

grams  of  water  formed  by  the 

melted  ice  from  0°  C.  to  5.5°  C.  =  110  x  5.5  calories. 
The  heat  required  to  melt  the  ice  =  110  x  calories. 
Hence,  110  x  +  110  x  5.5  =  300  (36.5  -  5.5)  ; 

whence,  x  =  79.9  calories. 

Careful  investigation  lias  shown  that  the  heat  of  fusion 
of  ice  is  80  calories. 

239.    Heat  Given  out  by  Freezing  Water.  —  According 
to  the  doctrine  of  the  Conservation  of  Energy  (§  64), 
heat   energy   equivalent   to   that   which    is    required   to 
change  a  solid  into  a  liquid  must  be  given  out  -when  the  , 
reverse  change  (i.e.  solidification)  takes  place. 

Make  a  freezing  mixture  of  snow  and  salt  stirred  well  together. 
Into  this  mixture,  which  will  be  several  degrees  below  0°  C.,  set  a 
test-tube  containing  water  and  a  thermometer.  If  the  water  in  the 
tube  is  not  disturbed,  it  may  reach  a  temperature  below  0°  C.  without 
freezing.  If,  however,  the  thermometer  be  moved  gently  against  the 
wall  of  the  test-tube,  the  water  quickly  begins  to  freeze  and  the  tem- 
perature rises  at  once  to  0°  C. 

In  the  freezing  mixture  of  snow  and  salt  used  in  the 
experiment,  both  solids  pass  into  the  liquid  state.  Just  as 
in  the  case  of  fusion  (§  238)  heat  is  absorbed  by  both  the 
salt  and  snow  during  the  change,.  The  heat  energy 
(kinetic  energy)  acquired  by  the  solids  in  liquefying  is 
converted  into  potential  energy  in  giving  their  molecules 
the  molecular  freedom  which  exists  in  a  liquid.  In  fact 


236 


A   HIGH   SCHOOL   COURSE   IN   PHYSICS 


the  solids  cannot  liquefy  unless  they  can  acquire  heat 
somewhere.  In  this  case  the  heat  is  taken  from  the  water 
in  the  test-tube,  and  thus  its  temperature  is  lowered  and 
a  portion  solidified.  On  the  other  hand  the  rise  in  tem- 
perature indicated  by  the  thermometer  which  was  placed 
in  the  test-tube  shows  that  heat  is  given  out  when  the 
water  changes  to  ice. 

When  a  gram  of  water  freezes,  80  calories  are  given  out  to 
its  surroundings.  The  enormous  amount  of  heat  evolved 
by  the  freezing  of  water  in  large  lakes  is  of  great  eco- 
nomic importance,  since  it  prevents  large  and  sudden  falls 
of  temperature  in  the  vicinity. 

EXERCISES 

1.  How  does  the  presence  of  tubs  of  water  in  a  cellar  tend  to  pre- 
vent the  freezing  of  vegetables? 

2.  What  has  the  large  heat  of  fusion  of  ice  to  do  with  the  rapid- 
ity with  which  snow  and  ice  disappear  on  a  warm  day? 

3.  In  freezing  cream  a  metal  vessel  B  (Fig.  177)  containing  it  is  sur- 

rounded by  a  rapidly  liquefying  mixture  of 
ice  and  salt,  A.  Give  a  complete  explanation 
of  the  process. 

4.  Can  a  piece  of  ice  be  warmed  above 
0°  C.  ?     Can  it  be  cooled  below  0°  C.  ? 

5.  Find  the  amount  of  heat  required  to 
melt  50  g.  of  ice  at  0°  C.  and  to  raise  the  tem- 
perature of  the  resulting  water  to  15°  C. 

6.  A  kilogram  of  ice  at  0°  C.  is  placed  in 
an  equal  mass  of  water  at  100°  C.     Find  the 


FIG.  177.— Liquid  B  Fro- 
zen by  the  Melting  of    resulting  temperature. 
Ice  in  A. 


Ans.  10°  C. 

7.  A  piece  of  ice  weighing  100  g.  and 
having  a  temperature  of  —  15°  C.  is  brought  into  a  room  where  the 
temperature  is  30°  C.  What  thermal  processes  take  place  ?  What 
quantity  of  heat  is  involved  in  each  process  ? 

8.   What  mass  of  ice  at  0°  C.  will  be  required  to  reduce  the  tem- 
perature of  a  kilogram  of  water  from  100°  C.  to  20°  C.  ? 

SUGGESTION.  —  Let  x  be  the  required  mass,  and  form  an  equation 
similar  to  that  used  in  §  238. 


HEAT:  TRANSFERENCE  AND   TRANSFORMATION       237 

240.  Evaporation  and  Ebullition.  —  We   have   seen   in 
§  235  that  the  change  from  the  solid  to  the  liquid  state 
takes  place  at  a  definite  temperature  in  crystalline   sub- 
stances.    The  change,  however,  from  the  liquid  to  the 
gaseous  state  occurs  at  all  temperatures  by  the  slow  pro- 
cess of  evaporation.     The  gas  that  rises  from  a  liquid  sub- 
stance is  called  the  vapor  of  that  substance.     Even  ice  and 
ice  water  evaporate.      But  by  sufficient  heating  a  liquid 
reaches  a  certain  temperature  at  which  the  familiar  pro- 
cess of  boiling,  or  ebullition,  begins.     This  temperature, 
which  is  called  the  boiling  point,  varies  greatly  with  dif- 
ferent substances  and  with  the  atmospheric  pressure. 

241.  Evaporation  Explained.  —  The  process  of  evapora- 
tion is  made  clear  by  the  help  of  the  molecular  theory 
(§  213).     At  the  exposed  surface  of  a  liquid  the  velocity 
of  many  of  the  molecules  is  sufficient  to  enable  them  to 
break  through  the  surface  beyond  the  range  of  attraction 
of   the  molecules  of  the  liquid.     If  the  temperature  be 
raised,  the  average  velocity  of  the  molecules  in  the  liquid 
is  increased,  and  a  more  rapid  surface  loss  will  result. 
The  removal  of  a  large  number  of  the  most  rapidly  mov- 
ing molecules  in  this  manner  decreases  the  average  kinetic 
energy  of  those  that  are  left  behind.     Consequently,  the 
temperature  of  the  remaining  liquid  is  lowered  by  the  pro- 
cess.   Evaporation  always  takes  place  at  the  expense  of  the 
heat  energy  contained  in  the  liquid. 

242.  Laws  of  Evaporation.  —  (1)    The  rate  of  evaporation 
becomes  greater  as  the  exposed  or  free  surface  of  the  liquid 
is  increased. 

A  pint  of  water,  for  example,  will  evaporate  faster  when 
placed  in  a  broad,  shallow  pan  than  when  left  standing  in 
a  pitcher.  When  spread  over  the  floor,  it  disappears  in  a 
short  time.  Hence  a  wet  cloth  will  dry  faster  when  spread 
out  than  when  left  folded. 


238          A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

(2 )  The  rate  of  evaporation  becomes  greater  as  the  tempera- 
ture of  the  liquid  and  vapor  is  increased. 

As  the  temperature  rises,  the  increased  molecular  motion 
enables  molecules  to  break  away  from  the  surface  at  a 
greater  rate. 

(3)  The  rate  of  evaporation  is  increased  by  the  removal  of 
the  vapor  from  the  space  above  the  liquid. 

When  the  space  above  the  liquid  contains  a  quantity  of 
the  vapor,  a  great  number  of  the  vapor  molecules  moving 
about  in  all  directions  by  chance  strike  the  surface  and 
reenter  the  liquid.  The  greater  the  number  of  molecules 
present  in  the  vapor,  the  larger  will  be  the  number  which 
return  to  the  liquid.  The  return  of  the  molecules  which 
have  once  detached  themselves  from  the  liquid  can  be  pre- 
vented by  removing  the  vapor  as  fast  as  it  is  formed. 
This  accounts  for  the  fact  that  roads  dry  quickly  on  windy 
days,  and  ink  is  frequently  evaporated  by  blowing  upon 
the  paper. 

243.  Vapor  Pressure.  —  When  a  liquid  is  placed  in  a 
vacuum,  it  rapidly  evaporates  until  a  condition  is  reached 
when  the  quantity  of  vapor  present  becomes  constant, 
i.e.  when  the  number  of  molecules  leaving  the  liquid  per 
second  is  just  equaled  by  the  number  which  reenter  it  in 
the  same  length  of  time.  The  vapor  is  then  said  to  be 
saturated.  A  vapor,  like  any  other  gas,  exerts  a  pressure 
against  the  walls  of  the  containing  vessel.  The  amount 
of  pressure  exerted  depends  upon  the  temperature ;  if 
the  temperature  rises,  more  of  the  liquid  evaporates  and 
the  pressure  increases;  if  the  temperature  falls,  some  of  the 
vapor  condenses  and  the  pressure  decreases.  The  pressure 
exerted  by  a  saturated  vapor  above  its  liquid  is  catted  the 
maximum  vapor  pressure  at  the  existing  temperature.  For 
example,  if  a  vessel  of  water  be  placed  in  a  vacuum,  it 


HEAT:  TRANSFERENCE   AND  TRANSFORMATION       239 

vaporizes  at  0°C.  until  the  vapor  pressure  is  4.6  millime- 
ters of  mercury;  at  10°  G.,  9.16  millimeters;  at  20°  C., 
17.39  millimeters,  etc. 

It  is  a  peculiar  fact  that  the  maximum  pressure  exerted 
by  a  particular  vapor  in  a  closed  space  is  independent  of 
the  pressure  of  other  vapors  that  may  be  present.  In  other 
words,  the  quantity  of  vapor  required  to  produce  saturation 
in  a  given  space  is  the  same  whether  that  space  is  a  vacuum 
at  the  beginning  or  is  occupied  by  other  vapors. 

244.  Unsaturated  Vapors.  —  When  the  vapor  present  in 
a  given  space  is  not  enough  to  produce  the  condition  of 
saturation,  i.e.  to  produce  the  maximum  vapor  pressure  at 
that  temperature,  the  vapor  is  called  unsaturated  vapor. 
This  will  always  be  the  case  in  a  closed  space  in  which  an 
insufficient  quantity  of  liquid  is  placed.      On  the  other 
hand,  when  a  vapor  is  kept  in  contact  with  its  liquid,  it  is 
always  saturated.     For  example,  the  vapor  above  a  liquid 
in  a  tightly  corked  bottle  is  saturated,  and  evaporation 
cannot    occur;   but  when  the  bottle  is  open,  the  vapor 
is  always  slightly  unsaturated,  and  therefore  a  continual 
change  of  the  liquid  into  a  vapor  takes  place. 

245.  Atmospheric  Humidity.  —  Since  evaporation  of  water 
is  always  taking  place  at  the  surface  of  lakes,  rivers,  and 
other  bodies  of  water,  and  also  from  the  soil  and  vegetation, 
there  is  always  more  or  less  water  vapor  present  in  the 
atmosphere.     That  this  is  the  case  may  be  shown  by  the 
following  experiment: 

Fill  a  polished  vessel  or  a  glass  beaker  with  ice  water,  and  allow  it 
to  stand  exposed  to  the  air.  In  a  short  time  drops  of  moisture  will  be 
seen  forming  on  the  exterior  surface  of  the  vessel. 

Ordinarily  the  air  does  not  contain  saturated  water 
vapor.  But  since  the  quantity  of  vapor  necessary  to  pro- 
duce the  condition  of  saturation  in  a  given  space  is  less  at 
low  temperatures,  the  air  in  contact  with  the  cold  vessel 


240          A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

soon  reaches  a  temperature  at  which  the  water  vapor 
already  present  becomes  saturated.  When  the  tempera- 
ture is  reduced  below  the  point  at  which  the  water  vapor 
is  saturated,  the  vapor  is  condensed,  and  moisture  is  de- 
posited upon  the  cold  surface  of  the  vessel.  The  tempera- 
ture at  which  moisture  begins  to  form  from  the  atmospheric 
water  vapor  is  called  the  dew-point. 

We  think  of  the  air  as  being  dry  or  moist  (i.e.  arid  or 
humid)  according  as  we  feel  that  it  contains  little  or  much 
water  vapor.  These  conditions  of  the  air,  however,  involve 
(1)  the  amount  of  vapor  actually  present  and  (2)  the  quantity 
necessary  to  produce  saturation  under  the  given  conditions. 
It  is  upon  the  relation  of  these  two  elements  that  the  sen- 
sations of  dry  ness  and  moisture  depend.  The  condition  of 
the  air,  in  regard  to  the  water  vapor  which  it  contains^  is 
expressed  by  the  ratio  of  the  mass  of  water  vapor  in  a  given 
volume  of  air  to  the  mass  of  vapor  required  to  produce  the 
condition  of  saturation  at  the  same  temperature*  This  ratio 
is  called  the  relative  humidity  of  the  air.  For  example,  if 
the  quantity  of  water  vapor  actually  present  in  a  given 
space  is  15  grams,  and  the  amount  required  to  produce 
saturation  at  that  temperature  is  20  grams,  the  relative 
humidity  is  f ,  or  75  %  - 

If  air  containing  water  vapor  is  caused  to  undergo  a 
decrease  of  temperature,  the  relative  humidity  increases 
since  the  cool  air  is  nearer  to  its  point  of  saturation.  If 
the  cooling  is  carried  far  enough,  moisture  which  we  call 
dew  is  deposited  on  solid  objects.  The  experiment  of  the 
beaker  of  ice  water  described  above  is  an  illustration  of 
this  effect.  If  the  temperature  at  which  the  moisture  is 
deposited  is  below  0°  C.,  it  is  frozen  as  fast  as  it  is  formed 
and  is  called  frost.  Similarly,  when  any  region  of  the  air 
cools  below  the  dew-point,  particles  of  water  produced  by 
slow  condensation  collect  about  dust  particles  and  produce 


HEAT:  TRANSFERENCE   AND  TRANSFORMATION      241 

fogs  and  mists.  When  a  condensation  takes  place  at  high 
altitudes,  clouds  are  formed.  The  slowly  falling  cloud 
particles  may  unite  to  produce  a  drop  of  rain.  In  cool 
seasons  condensation  may  take  place  at  temperatures 
below  the  freezing  point.  In  this  case  the  result  is  snow, 
sleet,  or  hail. 

246.  Ebullition. — It  has  already  (§  243)  been  stated 
that  the  saturation  pressure  (maximum  vapor  pressure)  of 
water  increases  with  the  temperature.     While  at  10°  C.  it 
is  only  9.16  millimeters  of  mercury,  at  90°  C.  it  is  525  milli- 
meters, and  at  100°  C.,  760  millimeters.     It  is  clear,  there- 
fore, that  at  some  definite  temperature  (viz.  100°  C.  for 
water)  the  maximum  vapor  pressure  must  be  equal  to  a 
pressure  of  one  atmosphere,  or  760  millimeters.     At  this 
temperature   the  average   speed  of  the  molecules  of  the 
liquid  becomes  so  great  as  to  render  the  cohesive  force  be- 
tween them  unable  longer  to  retain  them.     Hence  small 
groups   of   molecules   nearest  the   heated    areas    assume 
greatly  enlarged  volumes  (bubbles)  within  which  practi- 
cally no  cohesion  exists,  because  of  the  vastly  increased 
distance  between  the  particles.     Thus  at  this  temperature 
ebullition,  or  boiling,  takes  place.     This  phenomenon  is 
marked  by  the  formation  of  bubbles  of   saturated  vapor 
that  rise  to  the  surface  and  burst.     The  temperature  at 
which  this  condition  of  the  liquid  is  reached  is  the  boil- 
ing point  of  the  liquid. 

247.  Laws  of  Ebullition.  —  1.   Fit  a  2-hole  rubber  stopper  in  a 
test-tube.     Thrust  a  thermometer  through  one  of  the  holes  and  an  open 
glass  tube  through  the  other.     Place  a  small  quantity  of  sulphuric 
ether  in  the  test-tube,  and  hold  it  in  a  vessel  of  water  at  a  temperature 
of  about  70°  C.     Soon  the  ether  will  begin  to  boil,  and  the  thermome- 
ter will  indicate  a  steady  temperature  of  about  35°  C.     (CAUTION.  — 
On  account  of  the  high  inflammability  of  ether  vapor,  the  tube  con- 
taining it  should  not  be  brought  near  a  flame.)     Place  a  finger  over 
the  end  of  the  open  glass  tube,  and  thus  carefully  allow  the  pressure 

17 


242          A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

of  the  ether  vapor  to  increase.     The  boiling  point  will  he  observed  to 
rise  several  degrees. 

2.  Let  a  round-bottomed  flask  be  half  filled  with  water,  and  the 
water  boiled  for  two  or  three  minutes  to 
enable  the  steam  to  expel  the  air.  Close 
the  flask  with  a  rubber  stopper,  and  in- 
vert it  on  a  stand,  as  shown  in  Fig.  178. 
Although  the  temperature  of  the  water  • 
will  fall  rapidly,  the  water  can  be  made 
to  boil  vigorously  by  pouring  cold  water 
upon  the  flask. 

3.   Set  a  beaker  of  water  at  90°  C.  or 
less  under  the  receiver  of  an  air  pump 
and   begin    to   exhaust   the    air.     The 
FIG.  178.  —  Water  Boiling         water  will  boil  vigorously  as  long  as  the 
under  Reduced  Pressure.          pressure  is  kept  sufficiently  reduced. 

The  experiments  just  described  illustrate  the  following 
general  laws  of  ebullition : 

(1)  Every  liquid  has  its  own  boiling  point,  which  is  in- 
variable under  the  same  conditions. 

(2)  The  boiling  point  of  a    liquid  rises  or  falls  as  the 
pressure  upon  the  liquid  increases  or  decreases. 

The  cold  water  poured  upon  the  flask  containing  steam 
causes  a  portion  of  the  vapor  to  condense.  This  reduces 
the  pressure  within  the  flask,  thus  lowering  the  boiling 
point  to  the  temperature  of  the  water.  If  the  air  has  been 
very  thoroughly  expelled  from  the  flask,  the  water  may  be 
kept  boiling  until  it  is  scarcely  lukewarm. 

Because  of  the  decrease  of  atmospheric  pressure  with 
an  increase  of  altitude,  the  boiling  point  of  water  at 
Altman,  Colo.,  probably  the  highest  town  in  the  United 
States,  is  about  88. 5°  C.  Cooking  processes  at  such 
heights  are  frequently  accompanied  by  many  difficul- 
ties. In  a  steam  boiler,  however,  where  the  pressure  is 
125  pounds  per  square  inch,  the  boiling  point  of  water 
reaches  170°  C. 


HEAT:  TRANSFERENCE  AND  TRANSFORMATION   243 

Solid  substances  dissolved  in  a  liquid  raise  its  boiling 
point.  A  saturated  solution  of  common  salt  boils  at  about 
109°  C.  But  the  vapor  rising  from  boiling  brine  is  pure 
water  vapor  and  condenses  at  100°  C.  under  an  atmospheric 
pressure  of  760  millimeters. 


TABLE  OF  BOILING  POINTS 

Pressure  760  millimeters 


Ether. 35°    C. 

Chloroform       ....     61°    C. 
Alcohol  .  78.2°  C. 


Benzine 80°  C. 

Turpentine 160°  C. 

Mercury '   350°  C. 


248.  Distillation.  —  When  a  liquid  is  vaporized  in  one  vessel  and 
the  vapor  afterwards  condensed  in  anoth'er,  the  process  is  known  as 
distillation.     By  this   pro- 
cess pure  water  can  be  ob- 
tained from  water  contain- 
ing   dissolved    substances. 

and  other  foreign  matter. 
The  liquid  to  be  distilled  is 
placed  in  a  vessel  A,  Fig. 
179,  and  boiled.  The  va- 
por is  conducted  through 
the  tube  B,  which  is  sur- 
rounded by  a  larger  tube 
C  containing  a  stream  of 
cold  water.  The  vapor  is 
condensed  on  the  cold  walls  of  the  small  tube,  and  the  resulting  liquid 
runs  out  at  the  lower  end  into  the  vessel  D. 

If  two  liquids  are  mixed  together  and  heated  in  vessel  A,  the 
one  having  the  lower  boiling  point  will  be  vaporized  first.  Its  vapor 
can  be  condensed  and-  collected  in  a  separate  vessel  D.  Alcohol  is 
thus  separated  from  fermented  liquors,  and  gasoline  and  kerosene 
from  crude  petroleum. 

249.  Heat  of  Vaporization.  —Fill  a  glass  flask  about  half  full 
of  water,  and  place  it  over  a  flame  to  boil.     Suspend  one  thermometer 
in  the  liquid  and  another  a  little  above  it.     While  the  water  is  boiling, 
read  both  thermometers.     They  will  continue  to  read  practically  alike. 


FIG.  179.  —  Illustrating  the  Process  of 
Distillation. 


244 


A  HIGH   SCHOOL  COURSE   IN    PHYSICS 


Continue  to  apply  heat  until  it  is  evident  that  the  temperature  of  the 
water  and  steam  is  not  raised  above  the  boiling  point. 

The  experiment  shows  clearly  that  the  heat  applied  to 
the  flask  is  not  utilized  in  raising  the  temperature  of  the 
water  or  of  the  steam.  As  in  the  process  of  melting  a  solid 
(§  238),  heat  is  here  transformed  from  kinetic  into  poten- 
tial energy  while  changing  the  molecular  condition  of  the 
water.  For  every  gram  of  water  vaporized  a  definite 
quantity  of  heat  disappears.  The  amount  of  heat  required 
to  change  a  gram  of  any  liquid  at  its  boiling  point  into  vapor 
at  the  same  temperature  is  called  the  heat  of  vaporization 1  of 
that  liquid.  It  represents  the  work  that  has  to  be  done  in 
producing  a  separation  of  the  molecules  of  the  liquid 
against  their  mutual  attractions. 

When  condensation,  the  reverse  of  vaporization,  takes 
place,  an  amount  of  energy  equal  to  the  heat  of  vaporiza- 
tion is  given  up  by  the  condensing  vapor.  Thus  the 
quantity  of  heat  required  to  vaporize  a  certain  mass  of 
water  at  100°  C.  is  all  delivered  up 
by  the  steam  when  it  returns  to  the 
liquid  state. 

250.  Heat  of  Vaporization  of  Water 
Measured.  —  The  heat  of  vaporiza- 
tion of  water  can  be  readily  meas- 
ured by  allowing  a  known  mass  of 
steam  to  condense  in,  and  deliver 
its  heat  to,  a  known  mass  of  water. 

Allow  steam  from  a  flask  of  boiling 
water,  Fig.  ISO,  to  p'ass  through  a  tube  into 
a  beaker  containing,  say,  400  grams  of  cold 
water.  A  trap  T  should  be  introduced  in 


Water.  cold  water.     During  the  experiment  note 

1  Sometimes  called  "latent"  heat  of  vaporization. 


HEAT:  TRANSFERENCE   AND  TRANSFORMATION      245 

the  initial  and  final  temperatures  of  the  water.  From  5°  C.  to  35°  C. 
is  a  good  range  over  which  to  work.  The  mass  of  steam  condensed 
is  found  by  ascertaining  the  gain  in  the  mass  of  water  in  the  beaker 
during  the  process. 

For  example,  let  the  initial  temperature  of  the  water  in  an  experi- 
ment be  5.6°  C.,  the  final  temperature  35°  C.,  and  let  the  mass  of 
water  at  the  beginning  be  400  grams,  and  at  the  close  419.5  grams. 

Let  x  be  the  heat  of  vaporization  of  water. 

The  heat  gained  by  the  cold  water  =  400  (35  -  5.6)  calories. 

The  heat  lost  by  the  19.5  grams  of 

water  which  was  formed  by  the 

condensed  steam  on  cooling  from 

100°  C.  to  35°  C.  =  19.5  (100  -  35)  calories. 

The  heat  delivered  by  the  steam  at 

100°  C.  in  changing  to  water  at 

100°  C.  =  19.5  x  calories. 

Hence,  19.5  x  +  19.5  (100  -  35)  =  400  (35  -  5.6) ; 

whence,  x  —  538  calories. 

The  heat  of  vaporization  of  water  accepted  by  physicists 
is  536  calories." 

251.  Artificial  Ice.  —  Certain  substances  that  are  gases  under 
ordinary  conditions  of  temperature  and  pressure  become  liquids  when  the 
pressure  is  sufficiently  increased.  This  is  the  case  of  ammonia  gas,  the 
gas  that  is  given  off  from  common  aqua  ammonia.  The  pressure  re- 
quired to  liquefy  this  gas  under  ordinary  temperatures  is  about  10 
atmospheres,  or  150  pounds  per  square  inch.  On  the  other  hand, 
liquefied  ammonia  returns  to  the  gaseous  state  when  the  pressure  is 
reduced,  and  for  each  gram  that  vaporizes  a  quantity  of  heat  equal  to  its 
heat  of  vaporization  (§  249)  is  abstracted  from  its  surroundings.  Upon 
these  principles  is  based  the  operations  of  the  artificial-ice  machines 
in  common  use. 

An  artificial-ice  machine  consists  of  three  .essential  parts :  (1)  the 
compressor,  (2)  the  condenser,  and  (3)  the  evaporator.  See  Fig.  181. 
The  compressor  is  a  pump  which  is  run  by  an  engine  or  motor  whose 
function  is  to  force  ammonia  gas  under  a  pressure  of  about  10  atmos- 
pheres into  the  coils  of  the  condenser  C.  Here  the  gas  liquefies  and 
gives  up  heat  to  the  surrounding  water  which  carries  it  away.  From 
the  condenser  coils  the  liquefied  ammonia  passes  through  the  regulat- 
ing valve  V  into  the  coils  of  the  evaporator  E,  where  the  pressure  is 


246 


A  HIGH   SCHOOL  COURSE  IN   PHYSICS 


kept  below  two  atmospheres  by  the  continual  removal  of  ammonia  gas 
by  the  compressor.  The  rapid  vaporization  of  the  liquid  ammonia 
under  the  reduced  pressure  in  these  coils  causes  it  to  take  heat  from 
the  surrounding  brine.  By  this  abstraction  of  heat  the  temperature 
of  the  brine  is  reduced  to  a  point  several  degrees  below  the  freezing 
point  of  water. 


FIG.  181.  —  Artificial  Cooling  and  Ice-making  Apparatus. 

In  the  production  of  ice  the  evaporator  E  is  so  constructed  that 
metal  vats  of  pure  water  of  the  desired  size  can  be-  lowered  into  the 
cold  brine  and  left  until  frozen.  These  vats  are  then  withdrawn  from 
the  brine,  and  the  ice  removed  to  some  place  of  storage. 

The  artificial  cooling  of  storage  rooms  is  brought  about  by  cooling 
brine,  as  in  the  manufacture  of  ice.  From  the  evaporator  the  cold 
brine  is  forced  through  coils  of  pipe  placed  about  the  walls  of  the 
rooms  to  be  cooled.  Inasmuch  as  there  is  no  chance  for  the  ammonia 
to  escape,  it  can  be  used  repeatedly  with  very  little  loss.  The  pres- 
sures are  controlled  by  the  regulating  valve  and  may  be  read  at  any 
time  from  the  gauges  placed  as  shown  in  the  figure. 

EXERCISES 

1.  Explain  the  formation  of  moisture  on  the  interior  surface  of 
windows. 

2.  The  temperature  of  blades  of  grass  and  leaves  of  trees  falls 
rapidly  on  cloudless  evenings.     What  has  this  to  do  with  the  forma- 
tion of  dew  ? 

3.  Does  heating  the  air  in  a  room  remove  the  water  vapor  ?     Why 
is  the  air  in  an  artificially  heated  room  usually  "  dry  "  ? 

SUGGESTION.  —  It  is  shown  in  §  245  that  the  dryness  of  the  air  does 


HEAT:  TRANSFERENCE   AND  TRANSFORMATION      247 

not  depend  wholly  upon  the  water  vapor  present  in  a  given  space. 
The  student  should  try  to  write  out  in  full  the  entire  explanation. 

4.  Heat  a  beaker  of  water  over  a  flame,  and  observe  that  small 
bubbles  rise  to  the  surface  long  before  the  boiling  point  is  reached. 
Compare  this  with  the  phenomenon  of  boiling. 

5.  When  steam  is  allowed  to  flow  through  a  tube  into  cold  water, 
a  loud  sound  is  produced.     Explain. 

6.  What  becomes  of  the  cloud  that  one  sees  near  the  spout  of  a 
teakettle  ?     Is  it  steam  ? 

7.  Will  clothes  dry  more  quickly  on  a  still  or  a  windy  day? 
Why? 

8.  How  much  heat  is  required  to  raise  the  temperature  of  30  g. 
of  water  from  0°  C.  to  100°  C.  and  convert  it  into  steam? 

9.  If  50  g.  of  steam  at  100°  C.  change  into  water  at  40°  C.,  how 
much  heat  is  given  out  ? 

10.  If  the  heat  delivered  by   10  g.  of  steam   in  condensing  at 
100°  C.  and  cooling  down  to  0°  C.  were  all  applied  to  ice  at  0°  C.,  how 
many  grams  of  ice  would  be  melted?  Am.  79.5  g. 

11.  A  vessel  containing  600  g.  of  water  at  20°  C.  is  heated  until 
it  is  one  half  vaporized.     How  many  calories  have  been  received  ? 

12.  The  boiling  point  "of  water  falls  1  centigrade  degree  for  a  de- 
crease  in   pressure   of  2.7  cm.     Find   the  boiling    point  when   the 
barometer  reads  74.5  cm. 

13.  The  boiling  point  of  water  falls  1  centigrade  degree  for  an 
elevation  of  295  m.  above  sea  level.     Find  the  temperature  of  boiling 
water  at  Denver,  Colo.,  altitude  1600  m.  above  sea  level. 

14.  If  water  boils  at  85.5°  C.  at  the  top  of  Mont  Blanc,  what  is  the 
altitude? 

15.  If  20  g.  of  steam  at  100°  C.  are  passed  into  500  g.  of  water  at 
5°  C.,  what  will  be  the  resulting  temperature?  Ans.  29.2°  C. 

16.  How  much  heat  will  be  required  to  convert  150  g.  of  ice  at 
0°  C.  into  steam  at  100°  C.  ? 

2.    THE  TRANSFERENCE  OF  HEAT 

252.  Three  Modes  of  Heat  Transmission.  —  Heat  is  trans- 
ferred from  one  point  to  another  in  three  different  ways ; 
viz.  by  conduction,  convection,  and  radiation.  By  conduc- 
tion, heat  passes  through  the  metal  walls  of  a  stove  or  along 
a  metal  rod  from  heated  regions  toward  cold  ones.  By 


248          A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

convection,  the  currents  of  air  set  up  by  a  hot  stove  trans- 
fer heat  to  the  distant  parts  of  the  room.  By  radiation, 
energy  from  the  sun  reaches  the  earth ;  and,  by  this  pro- 
cess, the  hands  held  before  the  fire  in  an  open  grate  or 
near  a  hot  stove  become  warm. 

253.  Conduction.  —  The  conduction  of  heat  is  simply  the 
transference  of  molecular  motion  from  those  portions  of  a 
body  where  such  motion  is  greatest  to  those  where  the  mo- 
tion is  less,  i.e.  from  the  warmest  parts  of  a  body  to  colder 
parts,  without  producing  any  sensible  motion  in  the  inter- 
vening parts.     The  facility  with  which  the  conduction  of 
heat  takes  place  varies  widely  with  the  nature  of  sub- 
stances. 

Join  together  three  similar  bars  of  copper,  brass,  and  iron  as  shown 
in  Fig.  182,  and  heat  the  junction  in  a  flame  for  several  minutes.  By 
sliding  the  tip  of  a  sulphur  match  from  the 
cold  end  of  each  bar  toward  the  flame,  ascer- 
tain at  what  point  it  ignites.  In  this  manner 
it  may  be  shown  that  the  copper  has  conducted 
heat  the  farthest  from  the  source  and  is,  conse- 
quently, the  best  conductor.  The  iron  will 
prove  to  be  the  poorest  conductor  of  the  three 
metals. 

Make  tests  by  using  a  piece  of  glass  tubing, 
FIG.  182.— Testing  the    the  stem  of  a  clay  pipe,  a  piece  of  crayon,  etc. 
Conductivity  of    These  substances  will  be  found  to  conduct  very 
Metals"  poorly. 

254.  Liquids  and  Gases  Poor  Conductors  of  Heat.  —  The 

conductivity  of  a  liquid  can  be  tested  as  follows : 

Through  a  cork  fitted  in  the  neck  of  a  glass  funnel  pass  the  tube 
of  a  simple  air  thermometer  (Fig.  170),  as  shown  in  Fig.  183.  Fill  the 
funnel  with  water  until  the  bulb  is  covered  by  about  half  an  inch  of 
the  liquid.  Pour  about  a  spoonful  of  ether  upon  the  water  and  ignite 
it.  Although  the  flame  is  at  all  times  separated  from  the  bulb  of  the 
thermometer  by  only  a  thin  layer  of  water,  the  liquid  in  the  tube  will 
remain  stationary. 


HEAT:  TRANSFERENCE   AND  TRANSFORMATION      249 


The  experiment  shows  clearly  that  water  is  a  very  poor 
conductor  of  heat.     Its  conductivity  is  less  than 
of  copper.     In  general,  all  liquids, 
except   mercury  and  other   metals 
in  a  molten  state,  are  to  be  classed 
as  poor  conductors. 

Gases  have  a  lower  conductivity 
than  liquids.  For  this  reason  many 
substances,  as  wool,  which  inclose 
a  large  amount  of  air  are  poor  con- 
ductors and  are  therefore  used  ex- 
tensively in  the  manufacture  of 
winter  garments.  Such  articles  of 
clothing  owe  their  warmth  to  the 
fact  that  they  prevent  the  loss  of 

the  natural  heat  FIG.  183.— Testing  the  Con- 
oft  he  body.  ductivity  of  Water. 

Winter  wheat  and  fruit  trees  are  pro- 
tected in  a  similar  manner  by  a  de.ep 
covering  of  snow.  A  flannel  "  holder  " 
prevents  the  transference  of  heat  from 
the  flatiron  to  the  hand,  and  a  piece  of 
ice  wrapped  in  a  woolen  blanket  is 
shielded  from  the  heat  of  the  atmos- 
phere. ^ 

255.  Convection.  —  Heat  is  trans- 
ferred in  liquids  and  gases  by  the  pro- 
cess of  convection^  i.e.  by  a  general  mass 
movement  of  the  heated  portions  away 
from  the  source  of  heat. 


FIG.  184.— The  Con- 
vection of  Heat  by 
Water. 


1.  Pass  the  ends  of  a  glass  tube,  bent  as 
shown  in  Fig.  184,  through  a  rubber  stopper 
fitted  to  the  neck  of  a  bottle  from  which  the 
bottom  has  been  removed.  Fill  the  apparatus 


250          A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

with  water,  and  place  a  small  quantity  of  oak  sawdust  in  the  liquid  to 
serve  as  an  index  of  the  motion.  If  a  flame  is  moved  back  and  forth 

between  the  points  A  and  B,  it 
will  be  seen  that  a  current  is  set 
up  in  the  direction  from  A  to- 
ward B.  In  a  short  time  the  en- 
tire mass  of  water  in  the  reservoir 
C  will  become  hot. 

2.  Make  two  openings  in  the 
side  of  a  crayon  box,  as  shewn 
in  Fig.  185.  Set  a  short  candle 
over  one  hole,  leaving  an  opening 

into  the  box.     Set  a  lamp  chim- 
FIG.  185.  — Convection  Currents  in  Air. 

ney  over  each  hole  and  attach  it 

to  the  box  by  means  of  melted  candle  wax.  Light  the  candle  and  then 
hold  some  burning  paper  above  chimney  A .  It  will  be  observed  that  the 
flame  and  smoke  of  the  burning  paper  will  be  drawn  downward,  thus 
showing  the  direction  of  the  draft  of  air.  At  the  same  time  heat  is 
transferred  upward  from  chimney  B  by  the  convection  currents  of  the 
hot  gases. 

256.  Convection    Explained.  —  Convection    is    brought 
about  by  the  expansion  of  fluids  (i.e.  liquids  and  gases) 
when  heated.     When  a  portion  of  the  fluid  is  warmed,  its 
volume  increases,  thus  decreasing  the  density  of  the  fluid. 
This  portion  of  the  body  of  the  fluid  is  then  forced  upward 
in  a  manner  similar  to  that  in  which  a  submerged  piece  of 
cork  is  forced  up  by  water.     (See  §  121.)     As  the  heated 
portions  of  the  fluid  rise,  they  carry  their  heat  with  them, 
and  colder  portions  flow  in  to  replace  them.     Convection 
currents  are  applied  to  the  heating  of  buildings  with  hot 
air  and  hot  water  arid  to  mine  ventilation.     The  trade- 
winds  and  the  Gulf  Stream  are  convection  currents  of 
enormous  proportions. 

257.  A  Hot-air  Heating  System.  —  Figure  186  shows  the 
method  employed  in  heating  a  house  by  means  of  hot  air. 
A  furnace  is  placed  in  the  cellar  and  supplied  with  fresh 
air  through  the  duct  A  leading  to  the  heating  chamber  c,  c. 


HEAT:  TRANSFERENCE  AND  TRANSFORMATION  251 


Here  the  air  is  heated  and  thence  conducted  through  large 
pipes  to  the  various  rooms  of  the  building.  A  large  part 
of  the  air  thus  led 
into  the  rooms  finds 
an  outlet  around  the 
windows  and  doors. 
Sometimes  provision 
is  made  for  its  escape 
into  a  cold-air  flue 
leading  through  the 
roof.  A  cold-air  duct 
B  is  often  introduced 
for  the  purpose  of  re- 
conducting  the  par- 
tially cooled  air  to 
the  furnace,  where  it 
is  again  heated  and 
sent  into  the  rooms. 
For  good  ventilation, 
however,  an  abundant  supply  of  fresh  out-door  air  should 
be  admitted  to  the  system  at  A.  The  circulation  of  air 
is  indicated  by  the  arrows.  The  fire  in  the  fire-box  is 
controlled  by  dampers.  These  are  often  regulated  by 
chains  extending  to  a  room  above,  and  shown  here  by  the 
dotted  lines. 

258.  A  Hot- water  Heating  System.  — A  hot- water  system 
of  heating  depends  upon  convection  currents  produced  as 
shown  in  Experiment  1,  §  255.  Water  is  raised  nearly  to 
the  boiling  point  in  a  heater  H,  Fig.  187,  placed  in  the 
basement.  From  the  heater  it  is  conducted  through  pipes 
to  iron  radiators  R  placed  in  the  various  rooms  of  the 
building,  while  the  cooler  water  from  the  radiators  is  led 
in  return  pipes  back  to  the  heater.  Thus  a  continuous 
current  of  water  is  maintained  until  the  pipes  are  closed  by 


FIG.  186.  —  Hot-air  Heating  System. 


252 


A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


valves  placed  near  the  radiators.  On  account  of  the  large 
exposed  surface  in  each  radiator,  the  heat  emitted  by  the 

hot  water  is  transferred  to 
the  surrounding  air.  In  or- 
der to  prevent  the  radiation 
of  heat  from  the  conducting 
pipes,  a  thick  covering  of  as- 
bestos, a  very  poor  conductor 
of  heat,  is  frequently  pro- 
vided. This  plan  of  heating 
affords  no  means  of  ventila- 
tion. 

259.  Radiation.  -  -  When 
we  stand  before  a  hot  stove 
or  a  grate  full  of  glowing 
coals,  we  readily  perceive  that 
the  surface  of  the  body  near- 
est the  fire  is  rapidly  heated. 
FIG.  187.— Heating  a  House  by  However,  the  intervening  air 

Means  of  Hot  Water.  .  -.  ,  , 

is  not  warmed   as   it   would 

be  if  the  heat  passed  to  us  by  conduction  or  convection. 
Again,  a  room  is  frequently  heated  by  the  sun's  rays,  while 
the  glass  through  which  the  rays  pass  remains  cold. 
Plainly,  the  medium  which  transmits  the  heat  energy  in 
these  instances  is  not  the  air  nor  the  glass.  In  order  that 
the  phenomena  just  described  and  others  of  a  similar  na- 
ture may  admit  of  explanation,  it  is  assumed  by  physicists 
that  all  space  is  filled  with  an  exceedingly  light  medium 
called  the  ether.1  The  properties  of  the  ether  are  such  that 
transverse  wave  motions  are  transmitted  by  it  in  a  manner 
somewhat  similar  to  that  in  which  the  waves  produced  by 
a  falling  pebble  are  carried  along  upon  the  surface  of 

1  The  term  must  not  be  confused  with  "  ether,"  which  is  a  well-known 
liquid  and  is  used  in  several  experiments. 


HEAT:  TRANSFERENCE  AND  TRANSFORMATION   263 

water.  Energy  thus  transmitted  is  called  radiant  energy, 
and  the  process  is  called  radiation.  If  this  energy  affects 
the  sense  of  sight,  it  is  called  light.  When  it  falls  upon 
the  hands,  it  produces  warmth.  Radiant  energy  becomes 
real  heat  only  as  it  falls  upon  matter  which  is  capable  of  ab- 
sorbing it  and  converting  it  into  the  energy  of  molecular 
motion. 

260.  Absorption  of  Radiant  Energy.  —  The  ability  of  a 
body  to  radiate  energy  depends  both  upon  its  temperature 
and  the  nature  of  its  surface.  Smooth  and  highly  polished 
bodies  radiate  poorly,  while  rough,  black  bodies  radiate 
well.  On  the  other  hand,  bodies  differ  in  their  power  to 
absorb  radiant  energy.  Those  that  radiate  well  also  ab- 
sorb well. 

Nail  two  pieces  of  tin  A  and  jB,  Fig.  188,  to  a  block  of  wood  as 
shown.     Coat  the  interior  surface  of  B  with  lampblack  and  attach  a 
match  to  the  exterior  surface  of  each  with 
melted  paraffin.     Now  place  a  hot  iron  ball 
midway  between  the  plates.     In   a   moment 
the  wax  on  B  will  soften,  and  the  match  will 
fall.  A 

C: 

The  experiment  clearly  shows  that 
the  blackened  surface  B  absorJbs  the 
energy  radiated  by  the  ball  faster  than 

xi       £*    i_^  A         rr       •  r   1-1      i     FlG-     188. —  The     Black 

the  bright  one  A.  If  pieces  of  black  surface  B  Absorbs 
and  white  cloth  are  placed  upon  snow,  Heat  Better  than  the 

Polished  Surface  A. 

the  rapid  absorption  of  the  radiant- 
energy  of  sunlight  will  cause  the  black  body  to  melt 
its  way  into  the  snow.  Since  the  white  cloth  reflects 
and  transmits  the  greater  portion  of  the  energy,  little  re- 
mains to  be  converted  into  heat.  This  fact  accounts  for 
the  general  use  of  light-colored  clothing  in  summer  and, 
in  part,  for  the  stifling  heat  developed  in  attics  under  dark- 
colored  roofing. 


254          A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

261.  The   Radiometer.  —  This   interesting    instrument, 
Fig.  189,  was  invented  by  Sir  William  Crookes,1  of  Eng- 
land, in  1873.     It  is  used  to  detect  radiant 
energy.     The   instrument   consists   of   a 
glass   bulb  from  which  the  air  has  been 
almost  exhausted  and  within  which  four 
diamond-shaped  mica  vanes  are  delicately 
pivoted  on  light   cross  arms.      One  face 
of  each  vane  is  coated  with   lampblack. 
When   radiant   energy   falls    upon  these 
vanes,  a  rotation  is  produced. 

Since  the  blackened  faces  of  the  vanes 
FIG.  i89~^oke's  absorb  radiant  energy,  they  are  raised  to 
Radiometer.  a  higher  temperature  than  the  bright 
faces.  Thus  the  few  remaining  molecules  of  gas  in  the 
bulb  have  their  speed  greatly  quickened  as  they 
come  in  contact  with  the  black  surfaces,  and  hence  rebound 
from  these  faces  with  a  strong  reaction.  It  is  this  reaction 
that  causes  the  vanes  to  move.  The  speed  of  rotation 
depends  upon  the  intensity  of  the  radiation  falling  upon 
the  instrument. 

262.  Selective  Absorption  Of  Bodies.  —  Place  a  radiometer 
near  a  lighted  lamp  and  between  them  set  a  glass  beaker.     After  the 
rate  of  rotation  of  the  vanes  has  become  uniform,  fill  the  beaker  with 
water.     The  rate  of  rotation  will  be  greatly  diminished.     Repeat  the 
experiment,  but  fill  the  beaker  with  carbon  disulphide.     It  will  be 
found  to  have  little  effect  on  the  motion  of  the  vanes.     Substitute  a 
solution  of  iodine  in  carbon  disulphide.     Although  nearly  opaque  to 
light,  the  solution  will  be  found  to  transmit  the  radiations  perfectly. 

Water,  which  is  transparent  to  the  short,  or  visible, 
ether  waves  emitted  by  the  lamp,  transmits  very  poorly 
the  longer  waves  of  a  slower  rate  of  vibration.  Likewise, 
glass  transmits  well  the  visible  radiation  (i.e.  light)  from 
the  sun,  but  retards  effectively  the  longer  waves  emitted 
1  See  portrait  facing  page  466. 


HEAT:  TRANSFERENCE  AND  TRANSFORMATION   2,55 

by  the  objects  in  a  room.  Substances  like  glass  and  water 
which  absorb  long  waves  are  called  athermanous  substances. 
While  the  glass  in  a  window  admits  light  energy  into  a 
room,  the  energy  of  the  longer  waves  sent  out  from  the 
heated  objects  within  is  retained.  The  glass  of  a  green- 
house or  hot-bed  transmits  well  the  energy  of  short  waves 
to  the  soil  within,  but  the  longer  waves  emitted  by  the 
heated  soil  cannot  escape.  Hence  the  temperature  rapidly 
rises. 

On  the  other  hand,  the  carbon  disulphide  and  the  iodine 
solution  transmit  well  the  waves  of  the  ether  that  are  far 
too  long  to  affect  the  sense  of  sight.  Such  substances  are 
called  diathermanous  substances. 

263.  The  Sun  as  a  Source  of  Heat.  —  The  process  of 
radiation  plays  an  important  part  in  everyday  life.  The 
sun  is  continually  sending  out  great  quantities  of  radiant 
energy  in  all  directions  in  space.  A  small  fraction  of 
this  energy  falls  upon  the  earth's  atmosphere,  passes  read- 
ily through  it  without  producing  any  appreciable  change, 
and  reaches  the  earth's  surface.  Here  a  large  part  of  the 
energy  of  the  ether  waves  is  transformed  into  heat,  i.e.  is 
absorbed.  The  earth  also  radiates  heat;  but  being  of  a 
low  temperature,  the  waves  emitted  by  it  are  longer. 
Since  the  presence  of  water  vapor  in  the  atmosphere 
renders  it  athermanous,  the  radiation  of  energy  away 
from  the  earth  is  greatly  hindered. 

It  is  the  radiant  energy  from  the  sun  converted  into 
heat  that  evaporates  water,  resulting  in  the  production 
of  vapor  and  rain.  Rains  produce  the  flow  of  rivers  and 
thus  give  rise  to  the  energy  derived  from  waterfalls.  The 
wood  we  use  and  the  food  we  consume  owe  their  value  to 
the  energy  which  they  have  stored  up  within  them.  This 
they  derive  from  the  sunlight  and  warmth  in  which  they 
grow.  Coal  received  its  energy  from  the  plants  that 


256          A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

flourished  under  the  solar  radiation  of  past  ages.  It  is 
this  energy  that  we  utilize  in  warming  our  houses,  cook- 
ing our  food,  and  that  we  convert  into  mechanical  energy 
through  the  help  of  the  steam  engine  for  running  factories 
and  aiding  transportation  over  both  land  and  water. 


EXERCISES 

1.  In  the  construction  of  brick  and  cement  houses  the  walls  are 
often  made  hollow.     Why? 

2.  Why  does  a  piece  of  iron  feel  colder  than  a  piece  of  wood  when 
both  have,  the  same  temperature? 

3.  What  is  the  normal  temperature  of  the  blood  on  the  centigrade 
scale?  of  a  living  room? 

4.  Does  woolen  clothing  supply  the  heat  that  maintains  the  tem- 
perature of  the  body  ? 

5.  Explain  under  what  conditions  a  workman  might  be  led  to  wear 
woolen  garments  to  keep  himself  cool. 

6.  Why  are  coverings  of  sheets  of  paper  often  sufficient  to  prevent 
plants  from  freezing  on  frosty  nights? 

7.  Why  does  dew  seldom  form  on  cloudy  nights?    Why  are  frosts 
almost  entirely  prevented  by  the  presence  of  clouds? 

8.  In  a  fireless  cooker  a  kettle  containing  vegetables  that  have  been 
boiled  a  short  time  is  surrounded  by  wool,  felt,  etc.,  and  left  a  few 
hours  to  complete  the  process  of  cooking.     Explain. 

9.  The  poor  conductivity  of  glass  causes  a  tumbler  to  crack  when 
hot  water  is  poured  into  it.     Explain.     Why  does  not  a  thin  glass 
beaker  crack  from  the  same  cause  ? 


3.   RELATION   BETWEEN   HEAT   AND    WORK 

264.  Heat  and  Mechanical  Energy. —  The  experiments 
made  in  section  212  show  clearly  that  an  intimate  relation 
exists  between  heat  and  work.  The  production  of  heat  at 
the  expense  of  mechanical  energy  is  one  of  the  most  com- 
mon phenomena  of  nature.  In  fact,  the  energy  expended 
in  nearly  all  mechanical  processes  passes  finally  to  the  form 
of  heat.  An  inquiry  into  the  reverse  transformation  of 


COUNT    RUMFORD     (SIR    BENJAMIN    THOMPSON) 
(1753-1814) 

The  name  of  Rumforrl  is  prominent  among  the  early  physicists 
who  engaged  in  confuting  the  theory  that  heat  was  a  substance. 
His  attainments  as  a  soldier  and  public  benefactor,  however,  are  of 
no  less  interest. 

Thompson  was  born  on  a  farm  near  Woburn,  Massachusetts,  and 
as  a  young  man  was  employed  in  teaching  school.  Having  been 
made  a  major  in  the  local  militia  by  the  governor  of  New  Hamp- 
shire, he  became  the  object  of  mistrust  by  the  friends  of  American 
liberty.  On  this  account,  in  1776  he  removed  to  London.  Here  his 
advance  was  rapid;  within  four  years  he  became  undersecretary  of 
state.  In  1779  he  was  elected  a  member  of  the  Royal  Society.  In 
1783  he  planned  to  aid  the  Austrians  against  the  Turks,  but  while 
on  his  way  he  met  Prince  Maximilian  (afterwards  elector  of  Bava- 
ria), who  induced  him  to  enter  the  Bavarian  military  service.  For 
eleven  years  he  remained  at  Munich  as  minister  of  war,  minister 
of  police,  and  grand  chamberlain  to  the  elector.  He  reorganized 
the  Bavarian  army,  suppressed  begging,  provided  employment  for 
the  poor,  and  established  schools  for  the  industrial  classes. 

In  1791  Thompson  was  made  a  count  of  the  Holy  Roman  Empire 
and  took  the  name  of  Rumford.  In  1799  he  was  instrumental  in 
founding  the  Royal  Institution  of  London  and  selected  Sir  Humphry 
Davy  as  the  first  lecturer.  In  remembrance  of  the  help  that  he 
received  in  his  early  days  from  attending  some  lectures  by  Professor 
Winthrop  at  Harvard,  Thompson  later  gave  an  endowment  which 
founded  the  professorship  that  bears  his  name.  His  last  years  were 
passed  near  Paris,  where  he  died  in  1814.  His  tomb  is  at  Auteuil. 


PIEAT:  TRANSFERENCE   AND  TRANSFORMATION      257 


FIG.  190.  —  Water  raised 
by  Heat  Energy. 


energy,  i.e.  from  heat  into  mechanical  energy,  is  an  inter- 
esting consideration. 

Let  a  tube,  bent  as  shown  in  Fig.  190,  project  through  a  rubber 
stopper  into  a  flask  A  half  filled  with  water.  The  tube  should  extend 
nearly  to  the  bottom  of  the  vessel.  Heat  the 
water  over  a  burner,  and  the  water  will  be 
elevated  into  a  tumbler  at  B. 

In  this  experiment  work  is  per- 
formed .upon  the  water,  and  heat  is 
converted  into  potential  energy,  which 
is  stored  in  the  elevated  water. 

265.  Early  Historical  Experiments. 
—  In  the  early  development  of  the 
subject  scientists  looked  upon  heat  as 
a  kind  of  material  substance  called 

calbric.  An  important  step  toward  ascertaining  the  true 
nature  of  heat  is  found,  in  the  experiments  of  Count  Rum- 
ford.1  He  showed  that  one  horse  used  as  a  source  of  power 
could  develop  sufficient  heat  by  friction  to  raise  26.5  pounds 
of  water  from  the  freezing  to  the  boiling  point  in  2|  hours. 
About  this  time  Sir  Humphry  Davy  (1778-1829)  showed 
that  two  pieces  of  ice  kept  below  the  freezing  point  could  be 
melted  by  rubbing  them  together.  The  first,  however,  to 
establish  the  relation  between  heat  and  work  by  expressing 
one  in  terms  of  the  other  was  James  Prescott  Joule2 
of  Manchester,  England. 

266.  Joule's  Experiment.  —  The  method   employed  by 
Joule  to  ascertain  the  exact  relation  between  the  calorie 
and  the  unit  of  mechanical  energy  consisted  in  measuring 
the  heat  produced  by  a  definite  quantity  of  work.     The 
heat  under  consideration  was  produced  by  the  rotation  of 
paddles   in   a    vessel    of   water   C,  Fig.   191.     The  work 
done  upon  the  water  produced  heat  enough  to  raise  its 


1  See  portrait  facing  page  256. 
18 


2  See  portrait  facing  page  258. 


258 


A  HIGH   SCHOOL  COURSE   IN  PHYSICS 


\M 


FIG.  191.  —  Illustrating  Joule's  Method  for 
Determining  the  Mechanical  Equivalent 
of  Heat. 


temperature  an  appreciable  amount.     The  rise  in  tempera- 
ture  being  measured,  the  quantity  of  heat  developed  could 

be  computed  as  the 
product  of  the  mass  of 
water  and  its  change 
in  temperature  (§  231). 
The  paddles  were 
turned  by  two  weights 
IF,  IF,  attached  to  cords 
so  arranged  as  to  ro- 
tate the  main  shaft. 
The  work  performed 
by  the  paddles  could  be  computed  as  the  product  of  the 
weights  and  the  distance  through  which  they  descended 
(§  55).  Thus  the  work  corresponding  to  a  calorie  of  heat 
could  readily  be  determined. 

267.  The  Mechanical  Equivalent  of  Heat.  —  Joule's  ex- 
periments upon  the  relation  of  heat  and  work  extended 
over  more  than  one  half  his  life.     They  have  been  care- 
fully repeated   (1879)  by  Professor   Rowland  of   Johns 
Hopkins  University,  with  apparatus  of   more  refinement 
and  precision.    As  a  result  of  these  experiments  the  follow- 
ing value  of  the  calorie  is  generally  accepted  by  physicists: 

1  calorie  =  427  gram-meters,  or  41,900,000  ergs. 

This   result  is  known  as  the  mechanical  equivalent  of 
heat,  or  simply  Joule's  equivalent. 

268.  Conversion    of    Energy.  —  The    experiments    per- 
formed by  Rumford,  Joule,  Rowland,  and  others,  relative 
to  the  conversion  of  mechanical  energy  into  heat,  serve  to 
verify  the  principle  of  the  Conservation  of  Energy  first 
stated  in  §  64.     Ignorance  of  this  great  law  of  nature  has 
led  men  at  all  times,  and  even  in  this  enlightened  period, 
to  undertake  to  construct  ^devices  whereby  useful  work 


JAMES    PRESCOTT    JOULE    (1818-1889) 

The  attention  of  Joule  was  turned  at  an  early  age  in  the  direc- 
tion of  physics  and  chemistry  by  the  influence  of  his  teacher,  John 
Dalton,  the  chemist.  Under  his  tuition,  Joule  was  initiated  into 
mathematics  and  trained  in  the  art  of  experimentation.  His  renown 
rests  upon  the  thoroughness  with  which  he  established  the  doctrine 
of  the  Conservation  of  Energy  (§64)  upon  an  experimental  basis. 
His  epoch-making  experiments  were  made  to  determine  the  mechan- 
ical equivalent  of  heat,  which  is  recognized  as  one  of  the  most  im- 
portant physical  constants.  Measurable  quantities  of  energy  were 
expended  in  revolving  paddles  in  water,  mercury,  and  oil,  and  cast 
iron  disks  were  rotated  against  each  other  and  the  resulting  quantity 
of  heat  ascertained.  The  results  of  these  experiments  left  no  doubt 
that  the  amount  of  heat  produced  by  a  definite  quantity  of  mechan- 
ical work  is  fixed  and  invariable.  For  this  great  scientific  achieve- 
ment Joule  received  the  Royal  Medal  of  the  Royal  Society  of 
England  in  1852,  and  eight  years  later,  when  men  of  science  more 
fully  understood  the  value  of  the  discovery,  he  was  presented  with  the 
Copley  Medal. 

Joule  was  the  son  of  a  wealthy  brewer  of  Manchester,  England, 
at  which  place  he  carried  on  his  experiments.  Important  laws 
relating  to  the  heating  of  electrical  conductors  and  valuable  con- 
tributions to  the  subject  of  electro-magnetism  must  also  be  accred- 
ited to  him.  The  joule,  which  is  used  as  a  unit  of  energy,  has  been 
so  named  in  his  honor. 


HEAT:  TRANSFERENCE  AND  TRANSFORMATION   250 

can  be  obtained  without  the  expenditure  of  an  equivalent 
amount  of  energy  of  some  form.  Such  devices,  if  possible, 
would  supply  enough  energy  to  keep  their  parts  in  motion 
when  once  started,  and  are  therefore  called  perpetual-motion 
machines.  Any  attempt,  however,  to  secure  useful  energy 
from  the  wind,  sunlight,  ocean  waves,  tides,  etc.,  is  praise- 
worthy, and  should  be  encouraged.  To  some  extent,  such 
efforts  have  not  been  fruitless.  But  since  in  the  best  ma- 
chines that  can  be  made  some  friction  will  exist,  there  will 
be  a  continual  conversion  of  a  part  of  the  energy  supplied 
to  the  machine  into  heat.  If,  therefore,  energy  is  supplied 
only  in  starting  the  machine,  no  matter  how  it  is  con- 
structed, it  is  sure  to  come  to  rest.  Consequently,  a  ma- 
chine at  best  can  only  transfer  or  transform  the  energy 
with  which  it  is  supplied. 

The  transformation  of  mechanical  energy  into  heat  is 
easily  accomplished.  .For  example,  a  bullet  is  warmed  by 
its  impact  with  a  target,  mercury  can  be  warmed  by  being 
shaken  vigorously  in  a  bottle,  a  mass  of  shot  will  rise  in 
temperature  if  allowed  to  fall  several  feet,  a  button  grows 
hot  when  rubbed  upon  a  piece  of  flannel,  etc.  The  reverse 
transformation,  i.e.  from  heat  into  mechanical  energy,  is 
not,  however,  so  readily  effected.  The  process,  neverthe- 
less, is  accomplished  in  steam  and  gas  engines  by  making 
use  of  certain  properties  of  gases. 

269.  Gases  Heated  by  Compression  and  Cooled  by  Expan- 
sion. —  We  have  seen  in  §  212  that  the  energy  used  in  com- 
pressing a  gas  is  converted  into  heat.  On  the  other  hand, 
when  a  gas  is  allowed  to  expand  against  pressure  and  perform 
work,  heat  is  given  up  and  the  temperature  of  the  gas  falls. 
In  other  words,  molecular  energy  is  expended  by  the  gas 
when  it  does  work.  In  order  to  utilize  this  process  in 
converting  heat  into  useful  work,  the  gas  is  allowed  to  ex- 
pand in  a  cylinder  and  thus  move  a  piston  P,  Fig.  192. 


260 


A  HIGH   SCHOOL   COURSE   IN   PHYSICS 


It  is  clear  that,  in  the  case  illustrated,  the  gas  performs 
work  in  raising  the  weight  W  to  a  higher  position.     A 
modification  of  this  process  is  employed  in  all 
\w\         steam  and  gas  engines. 

270.  The  Reciprocating  Steam  Engine.  —  The 
essential  parts  of  an  ordinary  steam  engine  are 
the  cylinder,  the  piston,  and  the  slide-valve 
mechanism,  represented  diagram matically  in 
Fig.  193.  The  gas  employed  is  steam  gener- 
ated by  the  combustion  of  fuel.  A  to-and-fro 
motion  is  given  to  the  piston  P  by  the  force 


FIG.  192.—  By  exerted  by  the  steam  which  is  applied  to  its 
Piston  Pa  two  faces  alternately.     The  operation  is  as  fol- 

Gas   DoeS1 


Work  on 
the  Weight 
W. 


Steam  under  a  pressure  of  several  atmospheres  (i.e. 
100  to  250  pounds  per  square  inch)  enters  the  steam 
chest  S  from  the  boiler.  From  S  the  steam  finds  an  entrance  into  the 
cylinder  through  the  port  N  and  drives  the  piston  to  the  left,  forcing 
any  gas  that  may  be  contained  in  the  space  C" 
through  the  port  M  and  out  of  the  engine  through 


FIG.  193.  —  Section  of  the  Steam 
Engine. 


the  exhaust  pipe  E.  The  motion  of  the  piston  is  communicated  to  the 
main  shaft  A  through  the  connecting  rod  R  and  the  crank  D.  As  the 
piston  approaches  the  left  end  of  the  cylinder,  the  sliding  valve  V  is 
moved  to  the  right  by  the  eccentric  F  and  the  eccentric  rod  R',  thus 


HEAT:  TRANSFERENCE  AND  TRANSFORMATION   261 

admitting  "  live  "  steam  through  M  into  the  cylinder  chamber  C",  and 
opening  the  port  N  to  allow  the  expanded  steam  to  escape.  The 
piston  is  now  driven  back  to  the  right,  and  the  sliding  valve  V  is  forced 
back  to  its  former  position  just  before  the  piston  reaches  the  end  of 
its  stroke  as  at  first.  The  operations  just  described  are  then  repeated. 
The  shaft  of  the  engine  is  provided  with  a  heavy  fly  wheel  E  in  order 
to  maintain  uniformity  of  speed. 

In  the  so-called  non-condensing  or  high-pressure  engines  the  exhaust 
steam  escapes  through  the  exhaust  pipe  E  into  the  open  air.  The  pis- 
ton of  such  an  engine  therefore  moves  continually  in  opposition  to  the 
pressure  of  the  atmosphere.  This  disadvantage  is  partially  removed 
in  condensing  engines.  In  engines  of  this  type  the  exhaust  pipe  E 
conducts  the  exhaust  steam  to  a  condensing  chamber  in  which  a 
spray  of  cold  water  hastens  its  condensation.  By  the  aid  of  a  pump 
operated  by  the  engine,  the  water  together  with  the  condensed  steam 
is  removed  from  the  condensing  chamber,  leaving  a  back-pressure  of 
only  a  few  ounces  per  square  inch  instead  of  one  atmosphere.  Thus 
by  decreasing  the  back-pressure  against  the  piston,  a  larger  quantity  of 
useful  work  can  be  obtained  from  the  steam,  and  the  efficiency  of  the 
engine  correspondingly  increased.  Condensers  are  not  used  on  loco- 
motives (1)  because  of  the  large  supply  of  cold  water  necessary  and 
(2)  because  of  their  inconvenient  size. 

271.  The  Gas  Engine.  —  With  the  development  of  the  automo- 
bile has  come  that  of  the  gas  engine  as  a  source  of  power.  To  this 
class  of  machines  belong  all  engines  utilizing  an  explosive  mixture  of 
gases  as  the  working  agent,  such  as  air  and  illuminating  gas,  or  air 
and  gasoline  vapor.  The  operation  of  the  so-called  "  four-cycle  "  gas 
engine  in  common  use  is  shown  in  Fig.  194. 

C  is  a  cylinder  within  which  moves  the  piston  P.  The  piston  is 
connected  with  the  crank  B  by  means  of  the  rod  A.  Upon  the  crank 
shaft  D  is  mounted  a  heavy  fly  wheel  W,  which  is  set  in  motion  by 
the  hand  on  starting  the  engine.  When  the  piston  moves  downward 
to  the  position  shown  in  (1).  an  explosive  mixture  of  gas  and  air  is 
drawn  into  the  cylinder  through  the  inlet  valve  I.  As  the  motion 
continues,  the  piston  moves  upward  and  compresses  the  mixture  in 
the  top  of  the  cylinder,  as  shown  in  (2).  At  about  the  instant  the 
piston  reaches  the  highest  point,  as  in  (5),  an  electric  spark  at  the 
spark-plug  S  ignites  the  gas;  an  explosion  ensues,  with  the  produc- 
tion of  much  heat,  and  the  expanding  gases  exert  an  enormous  pres- 
sure on  the  top  of  the  piston.  This  forces  the  piston  violently  down- 


262 


A  HIGH   SCHOOL   COURSE   IN  PHYSICS 


ward,  giving  motion,  and  hence  kinetic  energy,  to  the  heavy  fly  wheel 
W.  On  the  next  upward  stroke  the  products  of  the  combustion  of  the 
gas  are  driven  out  of  the  cylinder  through  the  exhaust  valve,  as  shown 
in  (4).  This  valve  is  opened  automatically  at  the  proper  instant. 
The  piston  having  traversed  the  length  of  the  cylinder  four  times,  the 


FIG.  194. 


(2)  ('3) 

•  Operation  of  a  Four-cycle  Gas  Engine. 


initial  conditions  are  restored,  and  the  operations  are  repeated.  It  is 
obvious  that  the  fly  wheel  must  be  made  heavy,  since  the  energy  given 
it  during  the  third  stroke  of  the  piston  has  to  keep  the  engine  and  ma- 
chinery in  motion  with  almost  constant  speed  during  the  three  fol- 
lowing strokes  of  the  cycle. 

When  gasoline  is  used  as  fuel,  the  inlet  pipe  7  leads  from  the  "  car- 
buretor," into  which  the  liquid  enters  as  a  spray,  vaporizes,  and  is 
mixed  with  the  proper  amount  of  air.  In  order  to  prevent  undue 
heating  of  the  cylinder  and  piston,  a  current  of  water  is  kept  in  circu 
lation  through  cavities  cast  in  the  walls  of  the  cylinder.  Some  manu- 
facturers supply  so-called  "  air-cooled  "  engines,  the  cylinders  of  which 
are  cooled  by  the  circulation  of  air  about  their  exterior  surface.  In 
this  instance  the  cylinder  is  cast  with  numerous  projections  for  the 
purpose  of  increasing  as  much  as  possible  the  amount  of  radiating 
surface. 


HEAT:  TRANSFERENCE   AND  TRANSFORMATION      263 


On  account  of  the  lightness  and  compactness  of  the  engine,  and  the 
small  space  occupied  by  the  fuel,  gasoline  engines  are  extensively  used 
to  propel  automobiles,  in  which  motors  of  from  one  to  six  cylinders 
may  be  seen.  Such  engines  are  also  widely  used  in  launches,  dirigible 
balloons,  aeroplanes,  pumping  stations,  machine  shops,  factories,  etc., 
on  account  of  the  small  attention  required  in  their  operation. 

272.  The  Steam  Turbine.  —  In  the  common  reciprocating  form 
of  the  steam  engine  a  large  amount  of  energy  is  lost  in  stopping  and 
starting  the  piston  and  connecting  rods  at  the  end  of  each  stroke.  It 


(2) 


FIG.  195.  —Principle  of  the  Steam  Turbine. 

is  only  within  the  last  few  years  that  inventors  Have  succeeded  in 
'designing  efficient  engines  of  the  purely  rotary  type.  The  operation 
of  a  steam  turbine  is  as  follows : 

Steam  under  high  pressure  is  conducted  through  a  series  of  stationary 
jets  A,  (/),  Fig.  195,  arranged  in  a  circle,  which  directs  it  obliquely 
against  a  series  of  blades  B  which  are  attached  to  a  rotating  drum, 
called  the  rotor.  The  rotor  is  fastened  to  the  main  shaft  of  the  engine. 
By  the  impact  of  the  steam  these  movable  blades  are  impelled  in  an 
upward  direction  and  thus  produce  rotation.  (2),  Fig.  195,  shows  the 
arrangement  of  several  series  of  movable  and  stationary  blades  used 
in  the  more  powerful  turbines.  Each  movable  blade  is  a  curved  pro- 
jection attached  to  the  exterior  surface  of  the  rotor ;  each  stationary 
blade  is  fastened  to  the  interior  surface  of  the  metal  case  surrounding 
the  rotor.  The  concave  surfaces  of  the  two  sets  of  blades  are  turned 


264          A  HIGH   SCHOOL  COURSE   IN    PHYSICS 

opposite  to  each  other  as  shown.  The  number  of  series  used  will  vary 
largely  in  turbines  of  different  power.  After  the  steam  has  passed 
through  the  first  series  of  movable  blades  B,  a  series  of  blades  C, 
which  are  stationary,  serves  to  direct  it  at  the  proper  angle  against 
the  next  series  of  movable  blades  D,  and  so  on  through  the  entire 
turbine. 

The  steam  turbines  will  find  extensive  use  in  ocean-going  steamships 
on  account  of  the  fact  that  they  are  free  from  the  objectionable  vibra- 
tion that  always  accompanies  engines  of  the  reciprocating  form.  At 
the  present  time  they  are  replacing  the  ordinary  steam  engine  in  the 
generation  of  electrical  power,  and  in  a  few  years,  no  doubt,  will  be 
found  wherever  energy  is  to  be  derived  from  steam. 

EXERCISES 

1.  Explain  how  the  energy  contained  in  coal  can  be  utilized  in 
performii^g  work. 

2.  What  energy  other  than  that  of  coal  is  often  employed  in  run- 
ning factories,  etc.  ? 

3.  The  heat  developed  by  the  combustion  of  a  gram  of  coal  of  a 
certain  grade  is  5000  calories.     How  many  kilogram-meters  of  work 
could  be  done  if  all  the  energy  could  be  used  for  this  purpose  ? 

4.  If  all  the  potential  energy  stored  in  a  500-kilogram  mass  of 
rock  at  an  elevation  of  200  m.  were  converted  into  heat,  how  many 
calories  would  be  produced  ? 

5.  The  energy  of  a  falling  body  is  transformed  into  heat  when  it 
strikes.     Compute  the  number  of  calories  of  heat  produced  when  a 
10-gram  mass  of  iron  falls  25  m. 

SUGGESTION.  —  Compute  in  gram-meters  the  energy  of  the  given 
mass  at  an  elevation  of  25  m.,  then  reduce  to  calories. 

6.  A  steam  engine  raises  8000  four-pound  bricks  to  the  top  of  a 
building  75  ft.  high.     How  many  calories  of  heat  are  thus  expended  ? 
If  the  efficiency  of  the  engine  is  10  %  (§  100),  how  many  calories  must 
be  developed  by  the  combustion  of  the  coal  that  is  used? 

7.  How  high  could  100  g.  of  ice  be  elevated  by  the  amount  of  heat 
required  to  melt  the  same  amount,  if  all  the  heat  could  be  utilized  for 
that  purpose? 

8.  Show  that  the   energy  required  to  vaporize   1  g.  of  water  at 
100°  C.  is  equivalent  to  the  work  done  by  a  force  of  10  kg.  in  moving 
a  body  a  distance  of  22.89  m.  in  the  direction  of  the  force. 

9.  Show  that  it  requires  more  energy  to  raise  the  temperature  of 


HEAT:  TRANSFERENCE   AND  TRANSFORMATION      265 

100  g.  of  iron  from  0°  C.  to  100°  C.  than  to  elevate  a  weight  of  400  kg. 
through  a  height  of  1  m. 

10.  The  average  pressure  of  the  steam  in  the  cylinder  of  an  engine 
is  125  Ib.  to  the  square  inch  and  the  area  of  the  piston  is  50  sq.  in. 
Compute  the  work  done  by  the  steam  during  each  20-inch  stroke  of  the 
piston. 

11.  If  the  mechanical  equivalent  of  the  calorie  denned  in  §  231  is  3.1 
foot-pounds,  compute  the  heat  units  lost  by  the  steam  in  Exer.  10  at 
each  stroke  of  the  piston. 

12.  At  what  rate  does  a  2-horse-power  engine  consume  coal  when 
working  at  its  full  capacity,  if  its  efficiency  is  10  %  and  the  coal  pro- 
duces 5500  calories  per  grain  ? 

SUMMARY 

1.  The  state  of  a  body  depends  largely  on  its  temper- 
ature.    The  temperature  at  which  a  solid  changes  to   a 
liquid  is  its  melting  or  fusing  point.     Definite  laws  gov- 
ern the  fusion  of  all  crystalline  bodies  (§§  234  to  236). 

2.  The  process  of  liquefaction  is  accompanied  by  an 
absorption  of  heat.     Molecular  kinetic  energy  (heat)  is 
converted  into  molecular  potential  energy  in  the  process. 
The  heat  of  fusion  of  a  substance  is  the  number  of  calories 
per  gram  required  to  liquefy  it  without  changing  its  tem- 
perature.    The  heat  of  fusion  of  ice  is  80  calories.     Non- 
crystalline  substances  when  heated  pass  through  a  plastic 
state  and  have  no  definite  melting  point  (§§  237  and  238). 

3.  When  a  liquid  solidifies,  an  amount  of  heat  equal 
to  the  heat  of  fusion  is  given  up  for  each  gram  (§  239). 

4.  ^Evaporation  may  take   place   at   any  temperature. 
It  is  a  process  by  which  many  of  the  more  rapidly  moving 
molecules  of  a  body  become  detached  and  pass  into  the 
surrounding  space.     The  gas  composed  of  these  detached 
molecules  is  called  a  vapor  (§§  240  and  241) . 

5.  The  rate  of  evaporation  of  a  liquid  varies  with  the 
amount  of  exposed  surface  and  its  temperature,  and  is  de- 


266          A  HIGH   SCHOOL   COURSE   IN   PHYSICS 

creased  by  the  presence  of  the  vapor  of  the  liquid  in  the 
space  around  it  (§  242). 

6.  When  the  number  of  molecules  leaving  a  liquid  per 
second  equals  the  number  reentering  it,  the  vapor  is  satu- 
rated.    Its  pressure  at  this  time  is  called  the  maximum  va- 
por pressure  at  that  temperature  (§  243). 

7.  The  quantity  of  a  given  vapor  required  to  produce 
saturation  in  a  given  space  is  the  same  whether  the  space 
is  occupied  by  other  vapors  or  not.     This  quantity  de- 
pends, however,  on  the  temperature  (§§  243  and  244).   • 

8.  On  account  of  the  abundance  of  water,  the  atmos- 
phere always  contains   more   or   less  water  vapor.     The 
temperature  to  which  air  would  have  to  be  reduced   to 
cause  moisture  to  form  is  called  its  dew-point  (§  245). 

9.  The  relative  humidity  of  the  air  is  the  ratio  of  the 
amount  of  water  vapor  present  in  a  given  volume  to  the 
amount  required  to  produce  saturation  at  that  tempera- 
ture (§  245). 

10.  Every  liquid  has  its  own  boiling  point,  which  is  in- 
variable under  the  same  conditions.     This  point  rises  or 
falls  according  as  the  pressure  upon  the  liquid  is  increased 
or  decreased  (§§  246  and  247). 

11.  The  heat  of  vaporization  of  a  liquid  is  the  number 
of  calories  required  to  convert  1  g.  of  it  at  its  boiling  point 
into  vapor  at  the  same  temperature.     The  heat  of  vapori- 
zation of  water  is  536  calories.     This  amount  of  heat  is 
given  up  by  the  vapor  when  it  condenses  (§§  249  and  250). 

12.  Heat   is  transferred  by  conduction,  convection,  and 
radiation.     Substances  are  classed  as  good  and  poor  con- 
ductors of  heat.     In  general,  liquids  (except  liquid  metals) 
and  gases  are  poor  conductors  (§§  252  to  254). 

13.  Liquids* and  gases  transfer  heat  by  a  general  mass 


HEAT  :  TRANSFERENCE  AND  TRANSFORMATION  267 

movement  of  the  heated  portions  away  from  the  source  of 
heat,  i.e.  by  convection.  Convection  is  the  result  of  the 
expansion  which  accompanies  a  rise  in  temperature. 
Heating  systems  depend  on  the  convection  of  heat  by  air, 
steam,  or  hot  water  (§§  255  to  258). 

14.  Radiation  is  the  process  in  which  energy  is  trans- 
ferred by  ether  waves.     The  energy  of  these  waves  is  trans- 
formed into   heat  when  absorbed  by  bodies.     The  best 
radiators  and  absorbers  are  rough,  black  bodies.     The  sun 
is  our  great  source  of  energy.     This  energy  we  receive 
through  the  process  of  radiation  (§§  259  to  263). 

15.  Heat  may  be  made  to  perform  work.     One  calorie 
is  equivalent  to  427  gram-meters,  or  41,990,000  ergs.    This 
result  is  known  as  Joule's  equivalent,   or   the  mechanical 
equivalent  of  heat  (§§  264  to  267). 

16.  Heat  energy  is  transformed  into  mechanical  energy 
in  steam  and  gas  engines.     A  gas  does  work  in  expanding 
against  a  movable  piston.     As  a  result  of  the  work  done, 
the  temperature  of  the  working  gas  is  lowered  (§§  268 
to  272). 


CHAPTER   XIII 
LIGHT:    ITS   CHARACTERISTICS   AND  MEASUREMENT 

1.     NATURE   AND   PROPAGATION   OF   LIGHT 

273.  Meaning  of  the  Term  "Light."  -Just  as  sound  is 
defined  as  undulations  in  the  air,  or  some  other  medium, 
that  produce  the  sensation  which  we  call  "  sound,"  so  light, 
in  the  same  sense,  consists  of  undulations  or  waves  in  the 
ether  that  produce  the  sensation  which  we  often  call  by 
the  name  "light."    (See  §  259.)     Not  all  ether  waves  can 
be  regarded  as  light  waves,  since  not  all  affect  the  organ 
of   sight;    but  all  ether  waves,  from  the  longest   to  the 
shortest,  transfer  energy,  and  therefore  may  properly  be 
classed  as  carriers  of  radiant  energy. 

274.  The  Ether  and  Ether  Waves.  —  The   theory  that 
light  is  wave  motion  in  the  ether  was  advocated  by  the 
Dutch  physicist    Huyghens  (1629-1695)  in    1678.     The 
theory,  however,  was  not  well  established  until  the  begin- 
ning of  the  nineteenth  century,  when  the  experiments  by 
Thomas  Young  of  England  and  Fresnel  of  France  placed 
it  on  a  firm  basis.     Ether  fills  all  interstellar  space  as  well 
as  the  spaces  between  the  molecules  in  bodies  of  matter. 
The  ether  is  also  of  extreme  rareness,  or  tenuity,  since 
planets  passing  through  it  suffer  no  appreciable  retarda- 
tion in  their  orbits. 

Ether  waves  possess  several  well-known  characteristics. 
They  are  transverse  waves  and  are  propagated  with  a  defi- 
nite speed,  and  this  speed  becomes  less  when  they  pass 
through  matter  such  as  glass,  air,  water,  etc.  Ether 
waves  may  be  reflected,  transmitted,  bent  from  their 


LIGHT:  CHARACTERISTICS   AND  MEASUREMENT      269 

courses,  or  their  energy  may  be  transformed  into  other 
forms  than  radiant  energy. 

Ether  waves  that  produce  an  effect  upon  the  sense  of 
vision  vary  in  length  between  about  0.00004  and  0.00008 
centimeter.  Hence  our  sense  of  sight,  with  its  narrow  lim- 
itations, does  not  enable  us  to  perceive  directly  ether  waves 
which  are  shorter  than  the  former  of  these  two  numbers  or 
longer  than  the  latter. 

275.  Speed  of  Light.  —  An  achievement  of  great  scien- 
tific importance  was  the  discovery  that  light  travels  with  a 
definite  speed.  Previous  to  the  year  1676  it  was  supposed 
that  light  moved  infinitely  fast,  because  no  one  had  found 
a  way  to  measure  so  great  a  velocity.  In  that  year  the 
Danish  astronomer  Roemer  (1644—1710),  as  the  result  of 
several  months'  work  with  the  instruments  at  the  Observa- 
tory of  Paris,  correctly  inferred  that  the  time  required  for 
light  to  traverse  the-  diameter  of  the  earth's  orbit  (about 
186,000,000  miles)  was  almost  1000  seconds.  Roemer  was 
led  to  this  conclusion 
after  making  a  series 
of  observations  upon 
the  eclipses  of  one  of 
the  satellites  of  -the 
planet  Jupiter.  At 

each  revolution  of  the  ^ ^  ^ 

satellite  s,  Fig.   196, 
in    its    orbit    around 

Jupiter    J",     it    passes      FIG.  196.—  Roemer's  Method  for  Determining 

behind    that     planet  the  Speed  of  Light, 

into  its  shadow  and  becomes  invisible  from  the  earth  at 
E.  By  measuring  the  interval  between  two  successive 
eclipses  of  the  satellite  it  was  apparently  possible  to  pre- 
dict the  precise  time  of  each  eclipse  for  many  months  in 
advance.  But  when  the  earth  was  at  E* ',  on  the  opposite 


•Sk 


270         A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

side  of  the  sun,  the  time  of  each  eclipse  of  the  satellite  was 
found  to  be  about  1000  seconds  later  than  predicted.  In 
order  to  account  for  this  difference,  Roemer  advanced 
the  idea  that  this  interval  was  precisely  the  time  that 
light  requires  to  pass  over  the  diameter  of  the  earth's 
orbit.  On  this  assumption  the  speed  of  light  is  186,000,000 
-j-  1000,  or  186,000  miles  per  second. 

Recent  measurements  of  the  speed  of  light  by  different 
methods  continue  to  show  that  it  is  about  186,000  miles,  or 
300,000  kilometers,  per  second. 

EXERCISES 

1.  The  circumference  of  the  earth  is  about  25,000  mi.     How  many 
times  could  this  distance  be  traversed  by  light  in  a  second? 

2.  The  distance  of  the  north  star  from  the  eartli  is  so  great  that  it 
requires  about  43  yr.  for  its  light  to  reach  the  earth.     Express  the 
distance  in  miles. 

3.  How  many  minutes  are  required  for  light  to  reach  the  earth 
from  the  sun  ? 

4.  Since  the  sun  moves  (apparently)  through  360°  in  24  hr.,  over 
what  arc  will  it  move  while  a  light  wave  is  on  its  way  from  the  sun  to 
the  earth  ? 

2.   RECTILINEAR  PROPAGATION  OF   LIGHT 

276.  Light  Travels  in  Straight  Lines. —  Set  up  a  small  screen 
about  midway  between  a  candle  flame  and  the  wall  so  that  it  casts  a 
well-defined  shadow.  Mark  the  edge  of  the  shadow,  and  extinguish 
the  candle.  Stretch  a  cord  between  the  candle  wick  and  the  line  mark- 
ing the  edge  of  the  shadow,  and  it  will  be  found  to  graze  the  edge  of 
the  screen.  Since  the  cord  is  straight,  the  course  taken  by  the  light 
from  the  candle  to  the  wall  is  a  straight  line. 

Further  evidence  regarding  the  fact  that  light  follows 
straight  lines  may  be  obtained  by  observing  the  path  taken 
by  a  beam  of  light  as  it  enters  a  partially  darkened  room 
where  the  air  contains  dust.  We  unconsciously  utilize 
this  important  fact  in  many  ways.  In  order  that  we  may 
see  an  object,  light  must  come  from  that  object  to  the  eye  ; 


LIGHT:  CHARACTERISTICS  AND  MEASUREMENT      271 

and  we  always  assume  that  the  object  sending  the  light  to  us 
is  located  in  the  straight  line  which  marks  the  direction  of 
the  light  as  it  enters  the  eye.  The  marksman  trains  his  gun 
along  the  line  of  direction  of  the  light  which  comes  from 
the  object  he  wishes  to  hit,  and  the  carpenter  selects  a 
straight  piece  of  lumber  by  "  sighting  "  along  its  edge. 

We  shall  see  later,  however,  that  light  deviates  from  a 
straight  line  under  certain  conditions,  but  that  the  devia- 
tion is  ordinarily  inappreciable. 

277.  Shadows.  —  A.  shadow  is  a  space  from  which  the  light 
from  a  luminous  body  is  wholly  or  partially  excluded  by  an 
opaque  body.  The  nature  of  a  shadow  depends  both  upon 
the  form  of  the  opaque  body  and  upon  the  form  of  the 
source  of  light. 

1.  Hold  an  opaque  body,  as  a  book,  between  a  very  small  source 
of  light,  as  an  electric  arc  light,  and  a  white  wall  or  screen.     A  very 
sharply  outlined  shadow  will  be  produced  upon  the  screen  for  all  posi- 
tions of  the  opaque  body. 

2.  Place  two  electric  arc  lights  about  15  centimeters  apart  in  a 
line  parallel  to  a  screen  or  wall,  and  produce  a  shadow,  as  in  Experi- 
ment 1.     Two  portions  of  the  shadow  are  now  easily  distinguished, 
viz.  a  dark  central  part  and  a  partially  illuminated  area  just  outside. 
The  experiment  may  be  performed  with  a  single  gas  flame  or  by  using 
two  oil  lamps  placed  a  few  centimeters  apart. 

When  the  source  of  light  L,  Fig.  197,  is  small,  and  an 
opaque  body  AB  intercepts  the  light,  a  region  of  darkness 


FIG.  197.  —  Illustrating  the  Shadow  Cast  by  a  Sphere. 

ABCD  is  produced  behind  it -as  shown  by  the  shading. 
This  space  is  the  shadow  of  AB.     It  is  obvious  that  the 


272         A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

form  of  the  shadow  may  be  found  by  drawing  straight  lines 
from  L  just  touching  the  edge  of  the  object  AB.  If  AB 
is  a  sphere,  it  is  clear  that  the  form  of  the  shadow  will  be 
that  of  a  truncated  cone  whose  top  rests  against  the 
sphere. 

When  two   sources  of   light  L  and  L',  Fig.    198,  are 
used,  no  light  from  the  source  L  enters  the  region  CABD1, 


FIG.  198.  —  Showing  the  Production  of  Umbra  and  Penumbra. 

and  none  from  L'  enters  the  space  C'ABD.  Now  the 
space  CABD  lies  within  both  of  these  spaces  and  hence 
receives  no  light  from  either  source.  The  remaining  shaded 
portions  receive  light  from  one  or  the  other  of  the  sources. 
Furthermore,  if  other  luminous  points  exist  between  L  and 
L1 ',  the  space  CABD  will  receive  no  light  from  any  of  them. 
The  portion  of  a  shadow  that  is  wholly  dark  is  called  the 
umbra,  and  the  portions  that  are  only  partially  illuminated 
are  called  the  penumbra. 

278.  Eclipses  Produced  by  Shadows.  —  In  the  preceding 
section  it  was  seen  that  the  section  of  a  shadow  that  falls 
upon  a  wall  or  the  ground  will  have  a  distinct  outline 
only  when  the  source  of  light  is  small.  If,  therefore,  the 
source  of  light  is  the  sun,  we  find  no  sharply  defined 
shadows,  i.e.  every  shadow  is  surrounded  by  an  indistinct 
region  which  is  partially  illuminated. 

Let  the  sun,  Fig.  199,  be  the  source  of  light,  and  the 
earth  the  opaque  body.  By  drawing  lines  tangent  to 
both  sun  and  earth,  as  shown,  we  find  that  the  earth  casts 


LIGHT:  CHARACTERISTICS  AND  MEASUREMENT      273 


a  shadow  of  which  the  umbra  is  cone-shaped  and  has  its 
apex  at  A.  Surrounding  this  is  the  penumbra  whicli 
varies  from  total  dark- 
ness near  the  umbra  to 
practically  full  illumi- 
nation near  its  outer 
limits.  If  the  moon 

M  in  its  monthly,  re VO-      FIG.  199.  —  Showing  the  Moon  Eclipsed  by 

lution  about  the  earth,  Entering  the  Earth's  Umbra" 

in  the  orbit  shown  by  the  dotted  line,  passes  completely  into 
the  earth's  umbra,  it  receives  no  light  from  the  sun  and  is 
thus  eclipsed.  Since  we  see  the  moon  only  by  the  light 
which  it  reflects  from  the  sun  to  the  eye,  this  phenomenon 
constitutes  a  total  eclipse  of  the  moon.  But  if  only  a  por- 
tion of  the  moon  enters  the  earth's  umbra,  it  suffers  only 
a  partial  eclipse. 

279.  Pin-hole  Images.  —  Images  are  readily  produced  by  means 
of  small  apertures.  If  a  hole  2  or  3  millimeters  in  diameter  is  made 
in  the  window  shade  of  a  darkened  room,  images  of  trees,  clouds,  and 
other  outdoor  objects  will  be  produced  on  a  screen  held  a  short  dis- 
tance from  the  opening.  Each  dimension  of  an  image  is  proportional 
to  the  distance  from  the  aperture  to  the  screen,  but  the  larger  the  im- 
age, the  less  distinct  it  is  in 
every  detail,  since  the  light 
is  distributed  over  a  larger 
area. 

The  reason  for  the  for- 
mation of  images  in  this 
manner  is  made  clear  in 
Fig.  200.  CD  is  a  candle, 
A  an  opaque  piece  of  wood 
or  cardboard  having  a  small 
aperture  at  jET,  and  B  is  a 
white  screen.  Light  from 
the  tip  of  the  candle  C, 
FIG.  200.  — An  Image  Produced  by  a  Small  for  example,  falls  at  all 
Aperture.  points  on  A,  but  only  that 

19 


274         A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

falling  at  H  is  transmitted.  This  portion  follows  the  straight  line 
"CH  to  F.  Likewise,  only  light  from  D  can  fall  at  E  on  the  screen. 
Thus  the  portions  of  light  from  the  several  points  of  the  object  CD 
build  up  the  inverted  image  EF. 

Numerous  images  of  the  sun  may  often  be  observed  upon"  the  side- 
walk when  the  light  passes  through  the  small  openings  between  the 
leaves  of  a  tree.  These  images  assume  interesting,  crescent-shaped 
figures  during  a  partial  eclipse  of  the  sun. 

EXERCISES 

1.  Hold  a  book  in  direct  sunlight,  and   from  the  section  of  the 
shadow  that  is  outlined  upon  the  floor,  infer  whether  we  should  treat 
the  sun  as  a  point  source  of  light.     Describe  the  shadow. 

2.  Hold  a  ball  in  direct  sunlight  about  5  ft.  from  the  floor  or  wall, 
and  ascertain  whether  or  not  it  casts  a  distinct  shadow.     Do  the  same 
beneath  an  uncovered  electric  arc  light.     Draw  figures  to  illustrate  the 
difference  in  the  two  shadows. 

3.  Describe  the  shadow  cast  by  the  moon,    'in  what  direction  does 
its  umbra  point  ?    Does  its  umbra  ever  reach  the  earth  ? 

4.  When  the   moon  enters   the  earth's  umbra,  is  its  darkening 
gradual  or  sudden?    Explain. 

5.  If  the  earth  should  pass  into  the  moon's  umbra,  what  phenom- 
enon would  be  observed  by  a  person  standing  in  the  shadow  ?     Would 
any  of  the  sun  be  visible?     Would  any  of  the  sun  be  visible  to  a  per- 
son standing  in  the  moon's  penumbra? 

SUGGESTION.  —  Draw  a  figure  representing  sun,  moon,  and  earth 
in  such  a  position  that  the  umbra  of  the  moon  just  touches  the  earth. 

6.  If  the  sun's  rays  make  an  angle  of  45°  with  the  horizontal  plane, 
how  long  is  the  shadow  cast  on  level  ground  by  a  vertical  pole  50  ft. 
high? 

7.  A  vertical  rod  10  ft.  in  height  casts  a  shadow  12  ft.  long  on  a 
level  sidewalk.     How  tall  is  a  tree  whose  shadow  at  the  same  time  is 
72  ft.  in  length  ? 

8.  How  could  one  find  the  height  of  a  building  by  employing  the 
method  suggested  by  Exer.  7  ? 

3.    INTENSITY   AND    CANDLE    POWER   OF   LIGHTS 

280.  Intensity  of  Illumination.  —  If  one  realizes  that  the 
waves  of  light  that  are  sent  out  from  any  given  source 


LIGHT:  CHARACTERISTICS   AND  MEASUREMENT      2Y5 

spread  out  in  all  directions,  it  is  readily  inferred  that  the 
intensity  of  illumination  will  decrease  as  one  recedes  from 
the  luminous  body.  We  are  also  led  to  the  same  conclu- 
sion by  the  fact  that  we  decrease  the  distance  from  a  lamp 
to  a  printed  page  when  we  wish  to  increase  the  amount  of 
illumination.  The  exact  law  is  readily  shown  by  experi- 
ment. 

Cut  in  a  large  cardboard  screen  A,  Fig.  201,  an  aperture  just  2 
inches   square.      Place  the   screen  1  meter  from  a  point    source   of 


FIG.  201.  —  The  Intensity  of  Light  Varies  Inversely  as  the  Square 
of  the  Distance. 

light,  preferably  an  electric  arc.  Now  place  a  second  screen  B,  upon 
which  is  drawn  a  square  precisely  four  times  as  large  as  the  aperture 
in  A,  i.e.  4  inches  square,  2  meters  from  the  light.  The  light,  which 
at  a  distance  of  1  meter  falls  upon  an  area  A,  at  a  distance  of  2 
meters  is  found  to  cover  precisely  4  equal  areas.  Hence  each  area  at 
B  receives  only  one  fourth  as  much  light  as  a  similar  area  at  A. 
When  the  second  screen  is  carried  to  C,  a  distance  of  3  meters  from 
Lj  the  light  which  passes  through  A  illuminates  9  equal  areas  at  C. 
Hence  each  area  at  C  receives  only  one  ninth  as  much  light  as  an 
equal  area  at  A. 

It  is  now  plain  that  when  the  distance  from  a  source  of 
light  is  doubled,  the  intensity  of  illumination  is  divided 
by  4  ;  and  when  the  distance  is  made  three  times  as  great, 
the  intensity  of  illumination  is  J.  Hence  the  experiment 
leads  us  to  the  conclusion  that  the  intensity  of  illumina- 
tion is  inversely  proportional  to  the  square  of  the  distance 
from  the  source  of  light. 

281.  Candle  Power  of  Lights.  —  The  law  of  intensity 
shown  in  the  preceding  section  is  used  to  compare  the 


276          A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

illuminating  powers  of  two  sources  of  light.  If  the 
intensity  of  one  of  the  lights  is  known,  that  of  the  other 
can  be  found. 

Place  a  lighted  candle  A,  Fig.  202,  1  meter  from  a  paper  screen  S 
and  four  similar  candles  at  a  point  B,  the  same  distance  on  the  oppo- 
site side  of  S.  It  is  now  clear  that  the  side  of  the  screen  facing  the 
4  candles  receives  4  times  as  much  illumination  as  the  other.  But 
the  two  illuminations  may  be  equalized  by  moving  the  4  candles  to  a 
greater  distance.  If,  now,  a  drop  of  oil  or  candle  wax  is  placed  on  the 
paper  screen,  it  becomes  possible  to  ascertain  when  the  illuminations 
are  equal,  since  the  spot  will  look  alike  on  the  two  sides  when  viewed 

at  the  same   angle.     To  pro- 
duce   equal    illumination    on 
SA  the  two  sides  of  S  (i.e.  to  di- 

vide the  illumination  pro- 
duced by  the  stronger  light  by 
4),  it'  will  be  found  necessary 

to  move  the  4  candles  to  a  dis- 
FIG.  202.  —  Showing  a  Method  of  Meas- 
uring the  Candle  Power  of  Lights. 

screen.  Hence  the  light-pro- 
ducing powers  of  the  two  lights  are  directly  proportional  to  the  squares 
of  their  respective  distances  from  the  screen.  \ 

It  is  clear  that  this  method  may  be  employed  in  the 
comparison  of  the  light-emitting  powers  of  two  sources. 
The  process  is  to  set  the  lights  so  that  they  illuminate 
the  two  sides  of  a  screen  equally;  then  the  ratio  of  the 
squares  of  their  respective  distances  from  the  screen  ex- 
presses the  ratio  of  the  intensities  of  the  two  lights.  A  screen 
upon  which  is  an  oiled  spot  is  used  in  the  Bunsen  photo- 
meter for  the  measurement  of  the  power  of  lights. 

The  unit  used  in  the  measurement  of  the  power  of 
lights  is  called  a  candle  power  and  is  approximately  the 
power  of  a  sperm  candle  of  the  size  known  as  "sixes" 
(meaning  six  to  the  pound),  burning  120  grains  per  hour. 

The  candle  power  of  a  Welsbach  gas  lamp  consuming 
about  3  cubic  feet  of  gas  per  hour  is  from  50  to  100,  and 


LIGHT:  CHARACTERISTICS  AND  MEASUREMENT      277 

that  of  ordinary  open  gas  flames  is  from  15  to  25,  while  the 
consumption  of  gas  is  from  5  cubic  feet  per  hour  upward. 
The  incandescent  electric  lamps  containing  a  carbon  fila- 
ment in  most  common  use  are  of  16  candle  power,  but 
those  of  greater  power  can  be  procured. 

EXERCISES 

1.  A  2-candle-power  light  is  placed  1.5  m.  from  a  screen.     Where 
muvSt  an  8-candle-power  light  be  placed  to  produce  the  same  illumina- 
tion on  the  screen  ? 

2.  In  measuring  the  candle  power  of  an  electric  light  it  was  found 
that  a  4-candle-power  light  placed  2  in.  from  a  disk  produced  the  same 
illumination  as  the  electric  light  at  10  m.    Compute  the  power  of  the 
electric  light. 

3.  If  a  book  receives  ample  illumination  when  placed  10  ft.  from 
a  50-candle-power  lamp,  how  far  must  it  be  placed  from  a  light  of 
5-candle-power  to  be  equally  well  illuminated  ? 


SUMMARY ' 

1.  Light,  physically  speaking,  consists  of  ether  waves 
which  produce  the  sensation  called  light,  i.e.  which  excite 
the  optic  nerve  (§§  273  and  274). 

2.  The  speed  of  light  is  about  186,000  miles  (300,000 
km.)  per  second  (§  275). 

3.  Light  is  propagated  in  straight  lines  in  a  uniform 
medium.     This  fact  gives  rise  to  shadows,  eclipses,  pin- 
hole  images,  etc.  (§§  276  to  279). 

4.  When  a  luminous  body  is  of  appreciable  size,  the  shad- 
ows of  opaque  bodies  consist  of  two  parts,  the  umbra,  or  re- 
gion of  no  illumination,  and  the  penumbra,  or  partial  shadow. 

5.  The  intensity  of  illumination  is  inversely  proportional 
to  the  square  of  the  distance  from  a  source  of  light  (§  280). 

6.  The  illuminating  power  of  a  source  of  light  is  meas- 
ured in  terms  of  the  candle  power.     This   unit  is  about 
equal  to  the  power  of  the  ordinary  household  candle  (§  281). 


CHAPTER   XIV 

LIGHT:  REFLECTION  AND  REFRACTION 
1.     REFLECTION  OF  LIGHT 

282.  Reflection  and  Transmission.  —  It  is  a  familiar  fact 
that  a  piece  of  glass  both  reflects  and  transmits  light  ;  for 
we  frequently  see  the  bright  sunlight  reflected  by  the 
glass  of  a  window  when  we  are  outside,  although,  as  we 
know,  a  large  portion  of  the  light  is  transmitted  to  the 
interior  of  the  house. 

By  means  of  a  mounted  mirror  M,  Fig.  203,  reflect  a  bright  beam  of 
sunlight  upon  a  pane  of  glass  AB,  held  obliquely.  If  a  sheet  of  paper 

be  placed  behind  the  glass 
at  (7,  the  transmitted  light 
will  fall  upon  it  ;  if,  again, 
i*  ke  placed  i°  the  position 
A  it  will  be  brightly  il- 
luminated  by  reflected 


The  ordinary  mirror 

FIG.  203.  —  Reflection  and  Transmission  of     makes    US6    of    the   re- 
Light  by  a  Pane  of  Glass.  flection    of   light  from 

the  surface  of  the  opaque  film  of  mercury  that  covers  its 
back.     Polished  metals  are  often  excellent  reflectors. 

283.  The  Law  of  Reflection.  —  Every  one  is  accustomed 
to  the  manner  in  which  light  is  reflected  by  a  mirror  on 
account  of  the  many  purposes  which  it  serves  in  everyday 
life  ;  but  the  following  experiments  may  be  performed  in 
order  to  establish  the  law  which  ordinary  observation  does 
not  reveal  : 

278 


LIGHT:  REFLECTION   AND   REFRACTION        279 


1.    By  means  of  a  mirror  held  in  the  hand,  reflect  a  beam  of  sun- 
light in  various  directions,  and  observe  the  position  of  the  mirror  in 
each  case.     Attach  a  cardboard  index  so 
that  it  shall  be  perpendicular  to  the  mir- 
ror, and  observe  how  it  points  in  relation 
to  the  beam  of  light  before  and  after  its 
reflection.     It  will  be  found  that  the  in- 
dex always  points  in  a  direction  midway 
g  between      the 

direct  and  re- 
flected beams 
of  light,  as 
shown  in  Fig. 
204. 


Mirror 


FIG.  204.  —  Illustrating  the 
Reflection  of  Light  by  a 
Mirror. 


2.    Attach  a  block  of  wood  to  a  plane 
mirror,  and  set  it  upon  a  line  ruled  across 
a  large  sheet   of   paper.     Place   a    small 
FIG.  205.— The  Angle  of  Re-   candle  about  a  foot  from  the  mirror  at  C, 
flection  r  Equals  the  Angle    Fig.  205.     Now  place  the  eye  near  the  plane 
of  Incidence  i.  of  the  paper?  and  get  two  pins  in  line  with 

the  image  of  the  candle  wick  seen  in  the  mirror.  Draw  a  line,  as  BO, 
through  these  pins  to  the  mirror.  Draw  also  a  line,  as  CO,  from  the 
center  of  the  candle  to  the  point  where  the  first  line  intersects  the 
mirror.  Draw  the  line  OS  perpendicular  to  the  mirror  at  this  point. 
Angles  COS  and  BOS  will  be  found  to  be  equal. 

Now  part  of  the  light  from  the  candle  O  follows  line  CO 
to  the  mirror  and  line  OB  after  being  reflected.  Angle 
COS  is  called  the  angle  of  incidence,  and  angle  BOS  the 
angle  of  reflection.  In  every  case  it  will  be  found  that 
these  two  angles  are  equal.  Hence,  the  angle  of  reflection 
equals  the  angle  of  incidence.  It  is  to  be  observed  that  these 
two  angles  are  in  the  same  plane,  which  in  Experiment  2  is 
represented  by  the  plane  of  the  paper. 

284.  Diffused  or  Scattered  Light.  —  Objects  are  visible 
to  us  either  by  the  light  which  they  emit,  as  in  the  case  of 
the  sun,  a  candle,  or  a  live  coal,  or  by  the  light  which, 
after  falling  upon  them  from  some  luminous  body,  they 
scatter,  or  diffuse.  Most  objects,  unlike  a  smooth  piece  of 


280 


A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


glass,  reflect  light  in  many  directions.  Thus  the  sunlight 
which  falls  upon  the  snow  is  diffused  ;  but  when  it  falls 
upon  smooth  ice,  it  is  reflected  as  from  a  mirror.  This  is 
because  the  tiny  reflecting  surfaces  of  snow  lie  in  all  con- 


(•)  (2) 

FIG.  206.  —  Diffusion  Compared  with  Reflection  from  a  Smooth  Surface. 

ceivable  positions,  as  shown  in  (1),  Fig.  206,  while  those 

of  ice  all  lie  in  one  smooth  plane,  as  in  (2). 

By  the  help  of  the  light  which  objects  send  to  our  eyes, 

we  judge  of  their  distance,  form,  size,  color,  and  brilliancy. 

Leaves,  grass,  flowers,  etc.,  diffuse  in  every  direction  the 

sunlight  that  falls  upon  them.     The  moon  also  is  visible 

because  of  the  sunlight  diffused  from  its  illuminated  sur- 
face ;  and  we  are  often  able  to 
trace  the  dim  outline  of  the 
new  moon,  although  it  is  in 
shadow,  because  of  the  sunlight 
which  the  earth  diffuses  back 
upon  the  moon's  dark  area. 

285.  Image  of  a  Point  in  a 
Plane  Mirror.  —  It  was  found  in 
§  283  that  light  is  reflected  by 
a  plane  mirror  so  that  the  angle 

FIG.  207.  — Production  of  an  im-  of   reflection   is    equal    to    the 

age  by  a  Plane  Mirror.  angle  Qf  incidence.       HeriCC  the 

light   which   starts   from   the    point   A,    Fig.    207,   and 


LIGHT:  REFLECTION   AND   REFRACTION        281 


takes  the  direction  AB  is  reflected  by  the  mirror  MN 
in  the  direction  BC,  so  that  angle  OBD  equals  angle  ABD. 
All  other  rays  that  may  be  drawn  from  A  to  the  mirror 
are  reflected  in  the  same  manner  ;  and  when  the  eye  is 
placed  at  E  or  E',  the  reflected  rays  appear  to  come  from 
a  point  A'  behind  the  mirror. 

286.  Waves  and  "Rays."  -It  is  easy  to  conceive  of  a 
train  of  waves  moving  outward  from  the  point  A>  Fig. 
208,  and  striking  against  a  plane  mirror  MN.  The  waves 
are  sent  back  from  the  mirror  as  though  they  emanated 
from  the  point  A'  behind  the  mirror.  Hence,  to  an  eye 
placed  at  E  the  effect  is  just  the  same  as  though  A  were 
the  light-emitting  point.  It  is  obviously  more  convenient 
to  locate  the  image  of  a  point 
by  the  help  of  "rays,"  as  in 
Fig.  207,  rather  than  by  the  use 
of  waves,  as  in  Fig.  208.  How- 
ever,  it  should  always  be  remem- 
bered that  a  so-called  "  ray  "  of 
light  is  simply  a  symbol  used  to 
represent  the  direction  taken  by 
a  portion  of  a  wave.  Thus  that 

part   of   a   Wave    of   light   which 

starts  from  A  toward  B  in  Fig. 

207    follows 


FIQ 


Light  from  A  Seem  to  Come  to 

the  Eye  from  A'' 


L 


the    course   ABO. 
287.    Image  of  an  Object  in  a 
Plane    Mirror.  --  By    applying 
the  law  of  reflection  it  is  easy  to 
locate  the  image  produced  by  a 
plane   mirror.      Let   AB,    Fig. 
FIG.  209.—  Manner  of  Locating    209,  be  an  object  and  MN  the 
an  image  illustrated.  mirror.     Let  A  0  be  an  incident 

ray  from  A  drawn  perpendicular  to  the  mirror.     The  re- 
flected ray  will  take  the  direction  OA.     (Why  ?)     Let  AD 


282 


A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


be  another  incident  ray  from  A,  whose  direction  DE, 
after  being  reflected,  is  found  by  making  angle  EDF  equal 
angle  ADF.  The  image  of  A  lies  at  the  intersection  A1  of 
the  reflected  rays  CA  and  DE  produced  backward  behind 
the  mirror.  It  is  plain  from  the  equality  of  the  triangles 
ACD  and  A! CD  that  AC  equals  A'C;  i.e.  the  image  of  a 
point  is  as  far  behind  a  plane  mirror  as  the  point  itself  is 
in  front.  We  may  now  employ  this  fact  in  locating  the 
image  of  the  point  B  at  B1. 

288.  Seeing  the  Image.  —  Imagine  an  eye  to  be  placed 
at  E,  Fig.  209.  Light  enters  the  eye  from  the  mirror 
MNas  though  it  came  from  A1 ',  although  it  actually  comes 
from  A.  Similarly,  the  eye  receives  light  by  other  rays 
as  if  its  origin  were  at  J?',  whereas  it  is  really  at  B. 
TfrefeT'is  nothing  behind  the  mirror  that  concerns  our 
vision,  and  the  light  is  not  propagated  by  the  medium 
except  in  front  of  the  mirror.  The  image  is  called  a  vir- 
tual image  to  distinguish  it  from  the  real  images  formed  by 
small  apertures  (§  279)  and  in  other  cases,  to  be  studied 
later. 


FIG.  210.  —  A  Result  of  Double 
Reflection. 


FIG.  211.  —  Diagram.  Illustrating 
Double  Reflection. 


289.   Double  Reflection.  —  If  two  plane  mirrors  are  placed 
at  right  angles  to  each  other,  as  shown  in  Figs.  210  and 


LIGHT:  REFLECTION   AND   REFRACTION        283 

211,  it  is  clear  that  a  large  portion  of  the  light  emanating 
from  a  point  A  will  be  reflected  twice,  once  at  the  surface 
of  each  mirror.  Thus  the  ray  AB  is  reflected  from  B  to  0 
by  the  vertical  mirror  OM,  and  from  0  toward  D  by  the 
horizontal  mirror  ON.  Likewise,  the  ray  AE  takes  the 
course  AEFGr,  being  reflected  at  E  and  F.  Now  FG  and 
CD,  and  all  other  rays  that  have  undergone  double  re- 
flection, diverge  as  though  they  emanated  from  the  point 
A",  which  is  at  one  corner  of  the  rectangle  A'AA^A!11. 
It  is  therefore  evident  that  three  images  may  be  seen  by 
placing  the  eye  in  such  a  position  as  to  receive  light  that 
has  suffered  two  reflections  as  well  as  that  which  has  been 
reflected  but  once. 

EXERCISES 

1.  A  pole  is  inclined  at  an  angle  of  45°  to  the  surface  of  water  in  a 
quiet  pond.     Construct  the  image  of  the  pole  seen  in  the  water. 

2.  Look   at  your  image  in  a  mirror,  and  lift  your  right  hand. 
Which  hand  of  the  image  appears  to  be  lifted  ?     Is  the  image  direct 
or  reversed  ? 

3.  Set  a  candle  or  a  tumbler  on  a  horizontal  mirror,  and  observe 
the  position  of  the  image. 

4.  What  would  be  the  result  of  covering  the  mirror  OM  in  Fig. 
211  ?     How  could  the  formation  of  image  A '"  be  prevented  without 
interfering  with  image  A '  or  A  "  ? 

SUGGESTION.  —  Place  an  opaque  screen  so  that  no  light  can  be  re- 
flected twice.  Show  how  this  can  be  done.  s 

5.  A  man  approaches  a  plane  mirror  with  a  velocity  of  3  m.  per 
second.     How  rapidly  is  he  approaching  his  image  ? 

6.  Try  to  read  a  printed  page  by  looking  at  its  image  in  a  mirror. 
Write  your  name  backward  on  a  sheet  of  paper,  and  then  look  at  the 
image  of  the  writing  in  a  mirror.     What  eifect  is  produced  by  the 
mirror  in  each  case  ? 

7.  Find  by  construction  the  shortest  vertical  mirror  in  which  a 
man  6  ft.  tall  can  see  his  entire  image  when  standing  erect. 

SUGGESTION.  —  Diagram  the  case,  and  then  draw  lines  from  the 
man's  eye  to  the  highest  and  lowest  points  of  the  image.  Consider 
the  length  of  the  mirror  employed  between  these  two  limits. 


284          A   HIGH   SCHOOL  COURSE   IN   PHYSICS 

8.  Two  lines  AB  and  BC  make  an  angle  of  60°  with  each  other. 
Show  how  to  place  a  mirror  so  that  A  B  may  represent  an  incident 
ray  of  which  BC  is  the  reflected  ray. 

2.  REFLECTION  BY  CURVED  MIRRORS 

290.    Spherical    Mirrors.  —  A    spherical    mirror     is     a 
polished  or  silvered  portion  of  the  surface  of  a  sphere. 
If  the  side  of  the  surface  toward  the 
center  of  the  sphere  is  used  to  re- 
flect light,    see  Fig.  212,    the  mir- 
\°  ror  is  concave;  if  th£  outer  surface 
is  used,  the  mirror  is  convex.     The 
center  of  the  sjphere  O  is  the  center 
of  curvature,  and  the  radius  GO  of 

FIG.  21U.  —  Section  of  a 

Spherical  Mirror.          the  sphere  is  the  radius  of  curvature. 
The  line  of  symmetry  XO  is  called  flie  principal  axis. 

Let  direct  sunlight  fall  upon  a  concave  <mirror  parallel  to  the  prin- 
cipal axis,  and  hold  a  small  card  in  front  of,  the  mirror  to  receive  the 
reflected  light.  Move  the  card  back  and  forth  until  the  illuminated 
spot  is  as  small  as  possible.  Measure  the  distance  from  this  spot  to 
the  mirror.  If  a  piece  of  tissue  paper  be  held  at  the  spot  where  the 
reflected  light  is  concentrated,  it  will  probably  take  fire.  If  the  air  in 
front  of  the  mirror  be  filled  with  crayon  dust,  the  convergence  of  the 
reflected  light  is  easily  made  visible. 

Figure  213  shows  the  effect  produced  when  parallel  rays 
of  sifhlight  fall  upon  a  concave  mirror.     At  every  point 
011  the  mirror  light   is    re- 
flected according  to  the  law 
of   reflection    (§  283)  ;    but 
on  account  of  the  curvature 
of  the  mirror,  each  reflected 
ray  from  the  beam  of  par- 
allel rays   is    sent   through 

the  point  F,  which  is  therefore  called  the  principal  focus. 
When  the  energy  thus  concentrated  at  the  principal  focus 


LIGHT:  REFLECTION   AND   REFRACTION        285 

falls  upon  paper,  enough  of  the  energy  is  transformed  into 
heat  to  ignite  it.  The  principal  focus  is  located  midway 
between  the  center  of  curvature  and  the  mirror;  i.e.  OF 
eo^als  one  half  OC.  The  distance  OF  is  called -the  focal 
length  of  the  mirror  or  the  principal  focal  distance. 

291.  Convex  Mirrors.  —  Try  to  concentrate  direct  sunlight  by 
employing  a  convex  mirror  in  the  same  manner  as  the  concave  mirror. 
While  sunlight  is  falling  upon  the  mirror,  look  toward  its  convex 
surface  through  a  piece  of  black  glass.  An  exceedingly  bright  point 
will  be  seen  located  apparently  behind  the  mirror. 

Figure  214  illustrates  the  manner  in  which  a  convex 
mirror  reflects  the  parallel  rays  of  sunlight.  The  light  at 
every  point  follows  the  law  of  reflection  ;  but  on  account 
of  the  form  of  the  surface,  it 
diverges  as  though  it  came  I— 

from  the  point  .F  behind  the  ii 

mirror.     Of    course"  no    heat 


will  be  produced  at  the  point  ^^ 

F,  inasmuch  as  the  light   does  FIG.  214.^  Parallel  Rays  are  Made 

net  actually  pass  through  it. 

Since  F  is  only  an  apparent  meeting  point  or  focus  of  the 

reflected  rays,  it  is  called  a  virtual  or  unreal  focus.  The 

principal  focus  of  a  con- 
cave mirror,  ho\i*ever, 
is  a  real  focus.  See 
*.  ^l*-^ £  §288. 


To    locate    the  principal 
focus   of   a   convex    mirror, 
let  a  beam  of  sunlight  pass 
FIG.  215.— Determining  the  Focal  Length    through    a  round  hole  in   a 

piece  of  cardboard,  as  shown 

in  Fig.  215.  Around  this  aperture  draw  a  circle  whose  radius  is  just 
twice  that  of  the  aperture.  Let  the  beam  fall  upon  a  convex  mirror, 
and  then  move  the  mirror  back  and  forth  until  the  reflected  light 
just  covers  the  larger  circle.  Triangles  abc  and  coF  are  practically 


286 


A   HIGH   SCHOOL   COURSE   IN   PHYSICS 


FIG.  216. — Image  Formed  by 
a  Convex  Mirror. 


equal.     (Why  ?)     Hence  the  distance  from  the  cardboard  to  the  mir- 
ror oe  is  equal  to  oF,  the  focal  length  of  the  mirror. 

292.  Images  Formed  by  a  Convex  Mirror.  —  The  student 
is  probably  familiar  with  the  small  image  that  is  seen  as 

one  looks  at  the  polished  sur- 
face of  a  glass  or  metal  ball 
or  the  convex  side  of  the 
bowl  of  a  spoon.  The  experi- 
ment may  be  made  with  a 
lighted  candle,  as  shown  in 
Fig.  216.  The  image  will  ap- 
pear to  be  behind  the  mirror 
and  is  always  smaller  than  the 
object,  which  in  this  case  is 
the  candle.  Compare  with  Fig.  208. 

293.  Constructing  the  Image.  —  To  show  diagrammati- 
cally  how  an  image  is  produced  by  a  convex  mirror,  let 
the  arc  MN,  Fig.  217, 

whose  center  of  curva- 
ture is  at  (7,  represent 
a  convex  mirror.  Let 
the  arrow  AB  be  the 
object,  and  draw  the 
ray  AG-  parallel  to 
the  principal  axis  OX. 
The  reflected  ray  GD 
will  apparently  come 
from  F  (§  291),  which  is  midway  between  0  and  O.  Let 
a  second  ray  AH  be  drawn  from  A  along  a  radius  of  the 
mirror.  Since  this  ray  falls  perpendicularly  upon  the 
mirror,  it  will  be  reflected  back  along  the  same  line  HA. 
Now  let  the  reflected  rays  GrD  and  HA  be  produced  until 
they  intersect  at  some  point,  as  a  behind  the  mirror.  This 
locates  the  image  of  the  point  A.  The  image  of  the  point 


FIG.  217.  —  Locating  the  Image  of  AB  by 
Construction. 


LIGHT:  REFLECTION   AND   REFRACTION        287 


B  may  be  located  in  the  same  manner  at  b.  A  line  drawn 
from  a  to  b  represents,  therefore,  the  image  of  the  object 
AB.  Is  the  light  from  A  actually  focused  at  a  ?  Is  the 
image  therefore  real  or  virtual?  Is  it  erect  or  inverted? 
Is  it  larger  or  smaller  than  the  object?  Could  any  rays 
other  than  those  selected  be  employed  ?  How  would  you 
find  the  direction  of  any  other  reflected  ray  ?  (See  §  283.) 
294.  Images  Formed  by  a  Concave  Mirror.  —  The  nature 
of  the  images  produced  when  light  from  some  object,  as  a 
candle,  falls  upon  a  concave  mirror  is  readily  shown  by  a 
series  of  experiments.  Excellent  results  can  be  obtained 
by  using  a  mirror  whose  radius  of  curvature  is  20  inches 
or  more. 

1.   Let  a  lighted  candle  be  placed  before  a  concave  mirror  at  a 
distance  somewhat  greater  than  the  radius  of  curvature.     Place  a  small 


FIG.  218.  —  Production  of  a  Real  Image  by  a  Concave  Mirror. 

cardboard  screen  between  the  candle  and  the  mirror,  and  move  it 
back  and  forth  until  a  good  image  of  the  candle  appears  upon  it.  The 
image  will  be  found  between  the  principal  focus  and  center  of  curva- 
ture. 

This  image  differs  greatly  from  that  produced  by  a  con- 
vex mirror  in  that  it  can  be  caught  upon  a  screen.  We 
are  not  obliged  to  look  into  the  mirror  to  see  the  image, 
because  the  light  that  emanates  from  a  point  in  the  candle 
is  actually  reflected  to  a  corresponding  point  on  the  screen. 
Such  an  image  is  a  real  image.  The  experiment  plainly 


288         A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

shows  that  ivhen  the  object  is  beyond  the  center  of  curvature, 
the  image  is  between  the  center  and  the  principal  focus,  is 
real,  inverted,  and  smaller  than  the  object. 

2.  Let  the  candle  be  placed  at  any  point  between  the  center  of  curva- 
ture and  the  principal  focus  and  the  image  caught  upon  a  screen.    In 
this  case  the  screen  has  to  be  placed  beyond  the  center.    (See  Fig.  218.) 

Here  we  shall  readily  find  that  when  the  object  is  placed 
between  the  center  and  the  principal  focus,  the  image  is 
beyond  the  center,  is  real,  inverted,  and  larger  than  the  object. 

3.  Let   the    candle  be  placed  between  the  principal  focus  and   the 
mirror.     In  this  case,  in  order  to  locate  the  image,  direct  the  eye 
toward  the   mirror   in   such   a  manner  as   to  receive  some  of  the 
reflected  light..   An  erect  image  will  be  seen. 

When  the  object  is  between  the  principal  focus  and  the 
mirror,  the  image  is  behind  the  mirror,  is  virtual,  erect,  and 
larger  than  the  object. 

4.  Project  upon  a  screen  the  images  of  some  distant  object,  —  clouds, 
trees,  buildings,  etc.     In  all  cases  it  will  be  found  that  the  images  are 
small  and  lie  practically  midway  between  the  mirror  and  its  center  of 
curvature.     Parallel  rays  of  sunlight  are  also  focused  at  this  point. 

The  experiment  shows  that  the  image  of  an  object  at  a 
great  distance  lies  near  the  principal  focus,  is  real,  inverted, 
and  smaller  than  the  object  itself. 

295.  Construction  of  Images  Formed  by  Concave  Mirrors. 
—  We  have  seen  in  §  293  how  an  image  can  be  located 
by  geometrical  construction.  The  same  method  may  be 
applied  to  the  cases  arising  from  the  use  of  a  concave 
mirror. 

Case  I.  —  When  the  object  is  beyond  the  center  of  curva- 
ture. 

Let  MN,  Fig.  219,  be  a  concave  mirror  whose  center  is  C.  Let  the 
object  be  AB.  Locate  first  the  principal  focus  F  (§  290).  Now  let  a 
ray  AG  be  drawn  from  the  point  A  of  the  object  parallel  to  the 
principal  axis.  This  ray  will  be  reflected  through  the  point  F. 


LIGHT:  REFLECTION   AND   REFRACTION        289 

(Why?)   Let  a  second  ray  A  D  be  drawn  through  the  center  of  curva- 
ture C.     Since  this  ray  is  perpendicular  to  the  surface  of  the  mirror  at 

M 


FIG.  219.  — A  Real  Image  of  AB  is  Located  at  ab. 

Z),  the  reflected  ray  takes  the  direction  DC.  The  two  reflected  rays 
GF  and  DC  obviously  meet  at  the  point  a.  Could  other  incident  rays 
be  drawn  from  A  ?  How  could  their  direction  be  found  after  reflection  ? 
Where  would  they  meet  the  reflected  rays  already  drawn/?  Hence  the 
image  of  A  is  at  the  point  a.  In  a  similar  manner  the  image  of  the 
point  B  is  located  at  b.  Thus  ab  is  the  image  of  the  object  AB. 

When  two  points  -are  so  related  that  the  image  of  one 
falls  at  the  other,  as  A.  and  a,  or  B  and  £>,  they  are  called 
conjugate  foci  (pronounced /o' si). 

Case  II.  —  When  the  object  is  between  the  center  and  the 
principal  focus.  I 

The  conjugate  foci  of  the  points  A  and  B  are  located  by  a  method 
similar  to  that  used  in  the  preceding  case.  (See  Fig.  220.)  The  two 

AT 
b 


FIG.  220.  —  A  Real  Image  of  AB  is  Located  at  ab. 

cases  should  be  compared,  and  the  constructions  actually  made.     A 
real,  inverted,  and  magnified  image  is  found  at  ab. 
20 


290 


A  HIGH   SCHOOL   COURSE   IN   PHYSICS 


Case  HI.  —  When  the  object  is  between  the  principal  focus 
and  the  mirror. 

As  we  undertake  here  to  carry  out  the  method  of  construction  used 
in  the  preceding  cases,  we  find  that  the  reflected  rays  emanating  from 


FIG.  221. -  A  Virtual  Image  of  AB  is  Formed  at  ab. 


the  point  A  diverge  after  leaving  the  mirror.  (See  Fig.  221.)  This 
fact  shows  at  once  that  A  can  have  no  real  focus.  The  image  will  be 
seen  only  by  looking  into  the  mirror.  In  such  cases  the  reflected  rays 
DC  and  GF  are  to  be  produced  until  they  meet  at  some  point  as  a 
behind  the  mirror.  Similarly  the  image  of  B  is  found  at  6.  Thus 
a  magnified,  erect,  and  virtual  image  is  found  at  ab. 

Case  IV.  —  When  the  object  is  at  the  center  of  curvature  or 
at  the  principal  focus. 

When  light  from  a  point  at  the  center  of  curvature  falls  upon  a 
concave  mirror,  it  strikes  at  an  angle  of  90°  with  the  reflecting  sur- 
face and  is,  consequently,  reflected  back  along  the  same  path.  Hence 
all  such  rays  will  be  focused  at  the  center  of  curvature.  But,  when 
light  from  a  point  placed  at  the  principal  focus  is  reflected,  it  follows 
lines  parallel  to  the  principal  axis ;  e.g.  FGA  and  FHB,  Fig.  219. 
Since  such  rays  never  meet,  no  image  of  the  point  F  could  be 
produced. 

296.  A  Real  Image  Viewed  Without  a  Screen.  —  We  have 
already  seen  (§  294)  that  a  real  image  of  a  bright  object  can 
be  projected  by  a  concave  mirror  upon  a  suitable  screen. 
Now  if  the  eye  be  placed  about  10  inches  beyond  the  image 
and  turned  toward  the  mirror,  the  screen  may  be  removed, 


LIGHT:   REFLECTION   AND   REFRACTION        291 


and  the  image  will  be  visible.  Imagine  an  eye  at  JK, 
Fig.  222.  Light  waves  emanating  from  A,  a  point  in  the 
object,  advance  toward  the  mirror  with  convex  wave  fronts, 
as  those  of  water  waves 

s 

which  are  started  by  a 
falling  pebble.  These 
are  here  represented  by 
arcs,  having  their  com- 
inon  center  at  A.  But 
on  account  of  the  curva- 
ture of  the  mirror  MN, 
'the  waves  are  reflected 
with  concave  fronts 
having  as  their  common 
center  the  point  #,  which  is  called  the  conjugate  focus  of  A. 
As  the  waves  are  not  obstructed  by  a  screen,  they  leave  a 
with  convex  fronts  and  enter  the  eye  at  E. 


FIG.  222.  —  The  Eye  Can  Observe  a  Real 
Image  without  a  Screen. 


EXERCISES 

1.  An  object  is  placed  6  in.  in  front  of  a  convex  mirror  whose 
radius  of  curvature  is  12  in.     Find  by  construction  the  position  of 
the  image. 

SUGGESTION.  —  Make  a  drawing,  using  for  blackboard  work  the  di- 
mensions and  distances  given,  but  divide  each  by  4  for  pencil  drawings. 

2.  An  object  is  placed  44  cm.  from  a  concave  mirror  whose  radius 
of  curvature  is  50  cm.    Find  by  construction  the  location  of  the  image. 

3.  Place  a  small  object  slightly  above  the  center  of  curvature  of  a 
concave  mirror  whose  principal  axis  is  horizontal,  and  find  its  image 
by  construction.     Does  this  exercise  suggest  a  method  for  finding  the 
radius  of  curvature  by  experiment  ? 

4.  An  object  8  in.  in  height  is  placed  30  in.  in  front  of  a  concave 
mirror  whose  radius  of  curvature  is  15  in.     Find  the  distance  from 
the  mirror  to  the  image. 

5.  Make  an  accurate  construction  of  the  case  described  in  Exer.  4, 
and  carefully  measure  the  size  of  the  image.     Measure  also  the  dis- 
tances of  the  object  and  the  image  from  the  center  of  curvature.     Do 


292         A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

you  find  any  relation  between  the  distances  and  the  sizes  of  image  and 
object?  If  so,  express  it  in  a  single  sentence.  Can  you  prove  the 
same  relation  by  similar  triangles? 

3.    REFRACTION   OF   LIGHT 

297.  Refraction  of  Light.  —  Numerous  examples  of  the 
refraction  or  bending  of  the  course  taken  by  light  come 
before  our  attention  daily,  although  we  seldom  give  the 
phenomenon  much  thought.  A  simple  case  is  the  apparent 
bending  of  a  spoon  standing  in  a  tumbler  of  water,  or  an 
oar  at  the  point  where  it  enters  the  water.  Again,  if  a 
coin  be  placed  in  a  tumbler  of  water  and  viewed  obliquely, 
two  coins  become  visible,  —  a  small  one  seen  through  the 
horizontal  surface  of  the  water  and  a  magnified  one  seen 
through  the  side  of  tlie  vessel.  If,  now,  a  pencil  be  placed 
obliquely  in  the  water  contained  in  the  tumbler,  a  bent 
section  may  be  seen  below  the  upper  surface  of  the  liquid, 
while  a  magnified  portion  is  visible  through  the  side  of  the 
vessel.  In  every  case  the  illusion  is  due  to  the  bending  of 
light  rays  as  they  pass  from  one  medium  into  another.  No 
principles  of  optics  are  of  more  value  to  us  than  those  relat- 
ing to  the  phenomenon  of  refraction,  for  upon  this  effect 

,//,  are  based  not  only  our 
most  important  optical 
instruments,  including 
the  microscope,  tele- 
scope, and  camera,  but 
also  the  structure  of 
the  eye. 

298.    Refraction    II- 

FIG.  223.  -  Refraction  of  Light  as  it  lustr  ated.  —  1.  By  means 

Enters  Water. 

of  the  mirror  M,  Fig.  223, 

about  one  half  an  inch  in  width,  let  a  beam  of  sunlight  be  re- 
flected obliquely  upon  the  surface  of  water  in  a  tank.  By  scat- 
tering crayon  dust  in  the  air  above  the  water,  the  course  of  the 


LIGHT:  REFLECTION   AND   REFRACTION        293 

beam  before  and  after  entering  the  water  becomes  visible.  Another 
excellent  way  to  make  the  path  of  the  light  easy  to  trace  is  to  hold 
apiece  of  white  cardboard  or  tin  partly  under  water  so  that  it  receives 
the  beam  of  light  both  above  and  below  the  liquid  surface.  The  result, 
will  show  that  at  the  surface  of  the  liquid  O  the  beam  of  light  turns 
toward  the  perpendicular,  or  normal,  NN'  which  is  drawn  at  0.  If 
the  obliquity  of  the  incident  light  is  increased,  the  bending  of  the 
beam  is  made  more  pronounced. 

This  experiment  shows  that  when  light  passes  obliquely 
from  air  into  water,  it  undergoes  a  refraction  or  bending 
toward  the  perpendicular  to  the  surface  at  the  point  where  the 
beam  enters  the  water. 

2.  Place  a  plane  mirror  in  the  bottom  of  the  tank  used  in  the  pre- 
ceding experiment  so  as  to  reflect  a  beam  of  light  from  water  into  the 
air,  as  shown  in  Fig.  224.  The  course 
of  the  beam  MOP  may  be  traced  before 
and  after  entering  the  air  by  employing 
the  means  used  in  the  preceding  experi- 
ment. In  fact,  the  path  of  the  light 
MOP  is  precisely  the  reverse  of  that 
which  the  beam  would  take  if  it  were 
passing  into  water  along  the  line  PO. 

,.,       ,  T     .          FIG.    224.  —  Refraction    of 

It  will   readily  be  observed   in        Light  as  it  Emerges  from 
this    experiment    that   when   light        Water. 
passes   obliquely  from   water   into  air  it  undergoes  refrac- 
Uon  away  from  the  perpendicular  to  the  surface  at  the  point 
where  the  beam  emerges  from  the  water. 

The  angle  MON,  Fig.  223,  is  called  the  angle  of  inci- 
dence, and  angle  PON' ,  the  angle  of  refraction.  The  per- 
pendicular ON  is  usually  called  the  normal  at  the  point  0. 

299.  Cause  of  Refraction.  —  It  has  been  found  by  direct 
experimentation  that  light  waves  travel  with  less  speed  in 
water  than  in  air  ;  in  fact,  the  speed  of  light  in  water  is 
almost  exactly  three  fourths  of  that  in  air.  When  a  beam 
of  light  AB,  (1)  Fig.  225,  strikes  at  right  angles  to  the 
surface  of  water  (71),  all  parts  of  a  given  wave  front  strike 


294          A  HIGH   SCHOOL   COURSE   IN   PHYSICS 

the  medium  at  the  same  time.  Within  the  water  the 
waves  travel  with  less  speed  and  are  shorter.  Likewise,  if 
all  parts  of  the  wave  front  emerge  at  the  same  time,  they 
resume  their  original  speed  without  being  refracted.  But 
when  the  waves  fall  obliquely  upon  the  surface,  the  case  is 

quite  different.  See 
(2),  Fig.  225.  The 
parts  of  a  given  wave 
front  abed  do  not  enter 
the  medium  at  the 
same  instant;  but  a 
enters  first  and  con- 
(2)  tinues  with  reduced 

FIG.  225.—  Illustrating  the  Cause  sPeed'  while  the   other 

of  Refraction.  parts  bed  are  still  in  air. 

Similarly,  b  enters  the  medium  before  c  and  d,  then  c  be- 
fore d,  and  finally  d.  Thus  the  portion  a  travels  the  dis- 
tance act,',  while  d  is  traveling  the  larger  distance  dd' . 
The  result  is  that  the  wave  is  "  faced  "  in  a  different  di- 
rection, namely  aF,  having  suffered  a  bending  toward  the 
normal  to  the  surface,  NN1.  From  this  explanation  of  re- 
fraction it  is  clear  that  the  ratio  of  the  distance  ddr  to  the 
distance  aa'  is  the  same  as  the  ratio  of  the  speeds  of  liylit  in 
the  two  media. 

300.  Index  of  Refraction.  —  It  is  obvious  from  Fig.  223 
that  the  amount  which  the  course  of  light  is  changed  when 
it  enters  a  medium  where  its  speed  is  less  than  it  is  in  air 
depends  upon  the  relation  that  the  distance  aa'  bears  to 
the  distance  dd':,  or,  in  other  words,  upon  the  relation 
between  the  speeds  of  light  in  the  two  media.  Although 
it  is  not  easy  to  measure  the  speed  of  light  in  a  medium, 
it  is  comparatively  a  simple  matter  to  measure  the  amount 
of  refraction  and  from  this  to  compute  the  relative  speed 
of  light.  The  number  which  expresses  the  ratio  of  the  speed 


LIGHT:  REFLECTION   AND   REFRACTION        295 

of  light  in  air  to  its  speed  in  another  medium  is  called  the 
index  of  refraction  of  that  medium.     Hence 

speed  in  air 


Index  of  refraction^  = 


=  dd|    (l) 
' 


speed  in  other  medium     aa' 

301.  Index  of  Refraction  Measured.  —  By  referring  to 
Fig.  225  in  which  xx'  is  the  normal  at  the  point  d1  ',  we 
observe  that  angle  dad'  =  angle  dd'x,  which  is  the  angle 
of  incidence.  (Why?)  Angle  ad'  a'  =  angle  x'd't,  the 

angle  of  refraction.     (Why  ?)     Now  the  ratio  —  -  is  de- 

ad 

fined  in  mathematics  as  the  sine  of  angle  dad',  and  is  writ- 

ten sin  dad1.     Similarly,  the  ratio  —  is  the  sine  of  angle 

ad 

ad'a'  ,  and  is  written  sin  adfaf.     Hence 


sindd'x      sin  dadr      ad'     dd' 


aa 


°f  ref  ractlon' 


ad' 


Therefore  the  index  of  refraction  of   D 
a  substance  is  equal  to  the  quotient  ob- 
tained by  dividing  the  sine  of  the  angle 
of  incidence  by  the  sine  of  the  angle  of 
refraction. 

302.  Index  of  Refraction  by  Experi- 
ment. —  Let  a  glass  cube  A  BCD,  Fig.  226,  be 
placed  against  a  pin  P  set  upright  in  a  hori- 
zontal board  or  table.  Place  the  eye  at  some 
point  as  E,  and  set  two  other  pins  F  and  G  in 

Jine  with  P  as  seen  through  the  glass.  Draw  FlG.226.- Measuring  the 
line  AB  and  remove  the  cube.  Next  draw  index  of  Refraction  of 
lines  GFo  and  oP.  The  line  PoG  is  the  Glass. 

1  The  term  "  absolute  "  index  is  used  to  refer  to  the  ratio  of  the  speed 
of  light  in  a  vacuum  to  its  speed  in  a  medium.  The  index  of  refraction 
defined  above  is  often  called  the  "  relative  "  index. 


296          A  HIGH   SCHOOL   COURSE   IN   PHYSICS 

course  taken  by  the  light  that  enters  the  eye  from  the  pin  P.  Draw 
the  normal  bd  at  the  point  o,  and  then  draw  the  lines  ab  and  cd  per- 
pendicular to  the  normal  after  making  oa  =  oc.  Now  ab  H-  ao  is  the 
sine  of  angle  aob,  and  cd  •*•  co  is  the  sine  of  angle  cod.  Hence,  by 
equation  (2),  and  substituting  ao  for  its  equal,  co,  we  have 

ab 

.    ,         /•/...          ao      ab   m 
index  of  refraction  —  — :  =  — • 
J     J  cd      cd 


Therefore  we  can  readily  find  the  value  of  the  index  of  refraction  by 
dividing  the  length  ab  by  the  length  cd. 

RELATIVE  SPEED  OF  LIGHT  OR  ABSOLUTE  INDEXES  OF  REFRACTION 
FOR  SOME  COMMON  TRANSPARENT  SUBSTANCES 

Air 1.00029      Flint  glass     .     .     .     .  1.54  to  1.71 

Water 1.333          Carbon  disulphide      .     .     .    1.64 

Crown  glass      ....     1.51  Diamond 2.47 

303.  Total  Reflection.  —  Since  the  rays  of  light  which 
pass  from  water  or  glass  into  air  are  bent  away  from  the 

normal,  as  POA  and  PO'B,  Fig. 
227,  it  is  readily   observed    that 
Air         OsC/jo^tf"^        s    wnen  tne  angle  of  incidence  below 
water   /// \^^\  the  surface  is  great  enough,  the 

refracted  ray  will  follow  close  to 
the  surface,  as  ray  PO"S.     Hence 
the  ray  P  0"  is  the  last  one  that 
FIG.  227.  —  The  Total  Reflec-    can  emerge  from  the  surface.     If 

the  angle  of  incidence  'is  still  in- 
creased, the  light  is  reflected  wholly  beneath  the  surface, 
as  PO'"C.  This  phenomenon  is  called  total  reflection. 

1.  Examine  the  glass  cube  used  in  §  302  by  looking  through  the 
face  A  By  Fig.  226,  toward  face  BC,  which  has  the  appearance  of  a 
mirror.  Place  the  finger  upon  face  BC.  It  cannot  be  seen  through 
AB.  Transfer  the  finger  to  face  CD.  It  can  now  be  seen  in  face  BC 
by  reflected  light. 


LIGHT:  REFLECTION    AND   REFRACTION        297 

2.  Look  obliquely  upward  against  the  surface  of  water  in  a  tum- 
bler.    It  will  be  seen  to  have  the  appearance  of  a  plane  mirror.     If 
the  point  of  a  pencil  is  held  in  the  water,  only  that  part  of  it  is  visible 
that  projects  below  the  surface,  and  this  portion  can  also  be  seen  by 
reflected  light. 

3.  Reflect  a  narrow  beam  of  sunlight  obliquely  upward  through 
the  side  of  a  glass  tank  containing  water.     (See  Fig.  228.)     By  vary- 
ing the  angle  of  incidence  below  the 

surface  until  it  is  greater  than  48.5°, 
the  totally  reflected  beam  can  be  traced 
back  into  the  water.  Both  the  incident 
and  reflected  beams  may  be  made  vis- 
ible in  the  manner  described  in 
§298. 

4.  Hold  a  test-tube  obliquely  about 

5  centimeters  under  water,  and  look  vertically  downward  upon  it. 
The  portion  of  the  tube  below  the  liquid  surface  has  the  appearance 
of  a  mirror. 


FIG.  228.  —  An  Illustration  of 
Total  Reflection. 


These  experiments  serve  to  illustrate  the  fact  that  when 
the  angle  of  incidence  with  which  light  undertakes  to 
emerge  from  glass  or  water  exceeds  a  certain  value,  the 
light  is  totally  reflected  at  the  point  of  incidence  back  into  the 
medium.  The  angle  of  incidence  at  which  the  effect  changes 
from  refraction  to  total  reflection  is  called  the  critical  angle. 

The  phenomenon  of  total  re- 
flection occurs  only  when  light 
is  proceeding  in  one  medium 
toward  another  in  which  the 
speed  is  greater.  (See  Fig. 
229.)  The  critical  angle  for 
water  is  48. 5°;  for  crown  glass, 
41°;  for  diamond,  24°. 

FIG.  229.  — illustrating  the  304.    Critical  Angle  Con- 

Critical  Angle.  structed.  —  If  the  index  of  re- 

fraction of  a  medium  is  known,  the  critical  angle  can  be 
found  readily  from  the  following  construction : 


298 


A  HIGH   SCHOOL   COURSE   IN   PHYSICS 


Let  the  line  AB,  Fig.  230,  be  the  boundary  between  air  and  water, 
and  let  the  index  of  refraction  be  $ .     With  0,  the  point  of  incidence, 

as  the  center  draw  two  concentric  cir- 
cles whose  radii  have  the  ratio  of  4:3. 
Since  the  ray  that  emerges  for  the  crit- 
ical angle  follows  the  surface,  erect  the 
normal  at  the  point  (7,  where  the  sur- 
face intersects  the  inner  circle.  This 
cuts  the  larger  circle  at  E.  The  line 
EO  produced  below  the  surface  gives 
the  angle  DOF  as  the  critical  angle. 

The  proof  is  as  follows :  When  light 
of  Con-    passes  from  water  into  air,  the  index  of 
refraction  is  £ ;  and  for  the  critical  angle 
of  incidence  below  the  surface,  the  angle  of  refraction  is  90°.     It  is 
sin  DOF     3 
4 


FIG.    230.  —  Method 
structing  the  Critical  Angle. 


to  be  shown  that 


sin  90 


=  -..     Now,  angles  DOF  and    OEC  are 


equal. 

and  sin  90°=  1. 


(Why  ?)    Further,  sin  OEC  =  —  =  -  by  construction  (§  301), 

OE     4 


Therefore 
sin  DOF     sin  OEC 


sin  90° 


sin  90° 


305.  Total  Reflecting  Prism.  —  The  most  important  use  that  is 
made  of  the  phenomenon  of  total  reflection  is  accomplished  by  em- 
ploying a  right-angled  prism  of  glass  whose  cross 

section  is  an  isosceles  triangle,  as  shown  in  Fig. 
231.  If  incident  light  enters  the  prism  perpendic- 
ular to  either  of  the  faces  forming  the  right 
angle,  it  will  not  suffer  refraction  and  will  strike 
the  oblique  face  at  an  angle  of  incidence  of  45°. 
Since  the  critical  angle  for  glass  is  less  than  45°, 
the  light  will  be  totally  reflected  in  the  direction 
perpendicular  to  the  third  face,  from  which  it 
will  emerge  without  undergoing  refraction.  Such 
prisms  are  used  when  it  is  desired  to  turn  the 
course  of  light  through  an  angle  of  90°  without  excessive  loss. 

306.  Path  of  Light  through  Plates  and  Prisms.  —  The 

effect  of  a  parallel-sided  plate  of  glass  upon  a  ray  of  light 
is  readily  determined  by  the  following  experiment : 


FIG.  231.— Turning 
the  Course  of 
Light  through  90° 
hy  Total  Reflec- 
tion. 


LIGHT:   REFLECTION   AND   REFRACTION        299 

Look  obliquely  through  a  glass  cube  or  a  parallel-sided  tank  of 
water  and  set  four  pins,  two  on  each  side  of  the  cube  or  tank,  so  that 
they  will  form  apparently  a  straight  line.  Remove  the  refracting  ob- 
ject, and  draw  the  lines  connecting  the  two  pins  of  each  pair.  If  these 
two  lines  are  produced,  it  will  be  found  that  they  are  parallel. 

Hence,  when  light  passes  obliquely  through  a  medium 
with  parallel  faces,  it  does  not  suffer  a  permanent  change 
in  direction.  In  other  words,  the 
refraction  toward  the  normal  at  the 
first  surface  DO,  Fig.  232,  is  can- 
celed by  the  refraction  from  the  nor- 
mal at  the  second  surface  AB.  As 
the  experiment  shows,  the  ray  suf- 
fers a  lateral  displacement  only. 

The  course  taken  by  a  ray  of  light  FlGD  m  ~u^  Suffers  no 

•  *.  Permanent  Change  in  Di- 

whlCll   tails    obliquely    Upon   One    of         rection  in  Passing  through 

the  faces  of  a  glass  prism  may  be       Parallel  Surfaces- 
traced  in  a  manner  similar  to  that  just  described: 

Set  four  pins,  two  upon  each  of  the  opposite  sides  of  a  glass  prism, 
Fig.  233,  whose  angle  at  A  is  about  60°,  so  that  all  four  pins  lie 

apparently  in  a  straight 
line.  Draw  a  line  around 
the  prism  and  remove  it. 
Join  the  two  pins  in  each 
pair  by  a  straight  line,  and 
produce  these  lines  until 
they  meet  the  sides  of  the 
prism  AB  and  AC  &t  0 
and  0'.  Then  00'  is  the 

FIG.  233.  -  ^fraction  of  Light  by  Means        Path  of  the  HSht  throuSh 
of  a  Prism.  the  prisrn. 

Since  the  ray  which  enters  the  prism  is  bent  toward  the 
normal  at  0  and  away  from  the  normal  at  0?  where  it 
emerges,  it  is  clear  that  the  effect  of  the  prism  is  to  turn 
the  ray  always  away  from  the  refracting  angle  A  of  the 
prism. 


300          A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

EXERCISES 

1.  What  is  the  speed  of  light  in  water,  the  index  of  refraction  be- 
ing |?     The  speed  of  light  in  air  is  186,000  mi.  per  sec. 

2.  Compute  the  speed  of  light  ill  crown  glass,  assuming  that  the 
index  of  refraction  is  *. 

3.  Compare  the  speed  of  light  in  water  with  that  in  crown  glass. 
What  simple  fraction  will  represent  the  relative  speed?    Show  how  to 
get  this  same  fraction  from  the  indexes  of  refraction  given  in  Exer- 
cises 1  and  2. 

4.  The  answer  obtained  in  Exer.  3  is  called  the  relative  index  of 
refraction  for  light  on  passing  from  water  into  crown  glass.     In  a 
similar  manner  find  the  relative  index  of  refraction  for  light  which 
passes  from  flint  glass  into  carbon  disulphide,  using  the  indexes  given 
in  the  table. 

5.  For  a  given  angle  of  incidence  will  light  be  refracted  more  in 
passing  from  air  into  crown  glass  or  from  water  into  crown  glass? 
From   water  into  crown  glass  or  from  carbon  disulphide  into  flint 


6.  Construct  the  critical  angle  for  air  and  water.     On  which  side 
of  the  boundary  surface  does  the  critical  angle  lie  ?     Does  the  critical 
angle  lie  within  the  medium  where  the  speed  of  light  is  the  greater 
or  the  less  ? 

7.  Upon  which  side  of  the  boundary  surface  separating  water  and 
crown  glass  does  the  critical  angle  lie? 

8.  The  angle  of  incidence  at  which  a  ray  of  light  enters  a  medium 
from  air  is  45°,  and  the  angle  of  refraction  38°.     Find  by  construction 
the  index  of  refraction  of  the  medium. 

SUGGESTION.  —  By  the  help  of  a  protractor  construct  accurately 
a  figure.  Measure  the  lines  db  and  cd,  as  in  §  302,  and  compute  the 
index  of  refraction. 

9.  Draw  the  figures,  and  ascertain  the  indexes  of  refraction  for  the 
following  angles  :  angle  of  incidence  30°,  angle  of  refraction  22°;  angle 
of  incidence  50°,  angle  of  refraction  31° ;  angle  of  incidence  60°,  angle 
of  refraction  40°. 

4.   LENSES   AND   IMAGES 

307.  Lenses.  —  The  student  will  call  to  mind  many  in- 
struments with  which  he  is  acquainted  that  make  use  of 
of  lenses,  —  the  camera,  microscope,  spyglass,  spectacles, 


LIGHT:  REFLECTION   AND   REFRACTION        301 


opera  glass,  etc.  The  value  of  these  instruments  can 
hardly  be  too  highly  estimated.  The  study  of  lenses 
and  their  application  is  therefore  of  great  interest  and 
utility. 

A  lens  is  usually  made  of  glass  and  has  either  two 
curved  boundary  surfaces  or  one  curved  and  one  plane 
surface.  The  curved  surfaces  are  usually  (not  necessarily) 
spherical,  and  the  lens  thus  formed  is 
called  a  spherical  lens.  The  center  of 
the  sphere  (7,  Fig.  234,  of  which  the 
lens  surface  is  a  part,  is  called  the 
center  of  curvature,  and  the  radius  of 
the  sphere,  the  radius  of  curvature. 

Lenses  are  of  two  general  classes, 
—  convex  lenses,  which  are  thicker  at 
the  middle  than  at  the  edge,  and  con- 
caw  lenses,  which  are 'thickest  at  the  edge.  Figure  235 
shows  the  three  lenses  belonging  to  each  of  the  two  gen- 
eral classes. 


FIG.  234.  —  A  Spherical 
Lens. 


!5 


FIG.  235.  — Forms  of  Lenses. 

(1.  Double  Convex 
2.   Piano-Convex  Concave,  or  Diverg- 

3.   Concavo-Convex,  ing  Lenses 

or  a  Meniscus 


(4.  Doul 
5.  Plan 
6.  Com 


Double  Concave 

o-Concave 
Convexo-Concave 


308.   Effect  of  a  Convex  Lens  on  Light.  —  The  most  im- 
portant feature  of  any  lens  is  not  its  form,  but  the  manner 


302 


A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


in  which  it  acts  upon  light.  The  effect  of  a  lens  that 
is  most  familiar  to  all  is  that  employed  in  the  so-called 
"burning  glass,"  which  was  in  general  use  for  producing 
fire  before  the  introduction  of  cheap  matches.  The 
following  experiment  shows  this  effect : 

Allow  a  beam  of  sunlight  to  fall  upon  a  large  convex  lens  in  a 

darkened  room.  If  the  air 
be  made  dusty,  the  light 
will  be  seen  to  form  a  cone- 
shaped  figure  as  A  A,  Fig. 
236.  If  a  piece  of  tissue 
paper  be  placed  at  the  ver- 
tex of  the  cone,  it  is  readily 
ignited.  Beyond  this  point 

FIG.  236.  -  Effect  of  a  Convex  Lens  on         the  Hght  diverSes  Precisely 
Parallel  Rays.  as  from  the  principal  focus 

of  a  concave  mirror  (§  290). 

If  a  convex  lens  of  another  form  be  substituted  for  the  first,  the  action 
is  practically  the  same. 

The  vertex  of  the  cone  formed  by  the  sunlight  trans- 
mitted by  a  convex  lens  is  called  the  principal  focus  of  the 
lens.  The  distance  from  the  principal  focus  to  the  center 
of  the  lens  is  the  focal  length,  or  principal  focal  distance, 
of  the  lens.  Since  all  convex  lenses  cause  rays  of  sun- 
light to  converge  to  a  point,  they  are  often  called  converging 
lenses.  (See  Fig.  235.) 

Figure  237  shows  the 
change  in  form  that 
waves  of  light  undergo 
while  passing  through 
a  double  convex  lens. 
The  plane  waves  of 
sunlight,  whose  direc- 
tion of  motion  is  represented  by  arrows  drawn  in  the  figure, 
are  retarded  by  the  lens  in  proportion  to  the  thickness  of 


FIG.  237.  —  Plane  Waves  are  Converged  to 
the  Principal  Focus  F. 


LIGHT:  REFLECTION   AND   REFRACTION 


303 


the  glass  through  which  they  pass.  As  a  result  of  this  re- 
tardation, the  emerging  light  has  concave  wave  fronts 
whose  centers  are  at  F,  the  principal  focus.  On  leaving  F, 
however,  the  waves  have  convex  fronts  ;  or,  in  other  words, 
the  light  diverges  from  the  point  F. 

Ordinary  glass  lenses,  whose  index  of  refraction  is  very 
nearly  |,  have  a  focal  length  equal  to  the  radius  of  curva- 
ture, if  they  are  double-convex  and  the  two  surfaces  have 
the  same  curvature.  If  one  surface  is  a  plane,  the  focal 
length  is  double  the  radius  of  curvature. 

309.  Concave  Lenses.  — The  effect  of  a  concave  lens  on 
sunlight  is  very  different  from  that  of  a  convex  lens,  as 
the  following  experiment  will  show  : 

Let  a  beam  of  sunlight  fall  upon  a  concave  lens  mounted  in  a  wide 
board.  By  making  the  air  dusty,  or  holding  a  piece  of  cardboard 
obliquely  in  the  transmitted  light,  it  will  be  readily  observed  that 
the  light  diverges  as  it  leaves  the  lens. 

The  part  of  a  plane  wave  that  passes  through  the  center 
of  a  concave  lens,  Fig.  238,  is  retarded  least  on  account  of 
the  fact  that  the  lens  is 
thin  at  that  point,  while 
that  passing  through 
near  the  edge  suffers 
the  greatest  retarda- 
tion. The  result  is  that 
the  front  of  the  emerg- 
ing waves  is  convex,  as 
though  the  light  had 
emanated  from  the  point  F,  which  in  this  case  is  a  virtual 
focus  and  located  very  near  the  center  of  curvature.  Since 
the  general  effect  of  all  concave  lenses  is  to  cause  rays  of 
sunlight  to  diverge,  they  are  classed  together  as  diverging 
lenses.  (See  Fig.  235.) 


FIG.  238.  —Parallel  Rays  of  Light  are 
Scattered  by  Concave  Lenses. 


304 


A  HIGH   SCHOOL   COURSE   IN   PHYSICS 


310.  Conjugate  Foci.  — Place  a  candle  flame  close  to  a  small 
hole  P  in  a  piece  of  tin  S,  Fig.  239.  Now  set  a  convex  lens  about 
twice  its  focal  length  away  from  the  hole  P,  and  place  a  screen  S' 
on  the  opposite  side  of  the  lens  upon  which  will  be  produced  a  sharp 


FIG.  239.  —The  Conjugate  Focus  of  P  is  at  P'. 

image  of  P  at  P'.  Cover  up  one  half  the  surface  of  the  lens,  and  the 
image  will  remain  at  P'.  Cover  the  lens  almost  entirely,  and  the  image 
will  be  weakened  but  not  destroyed. 

It  is  to  be  observed  that  all  the  light  emanating  from 
the  point  P  and  passing  through  the  lens  is  collected  at 
P' .  Likewise,  any  other  point  beyond  the  principal  focus 
F  has  a  corresponding  point  on  the  opposite  side  of  the 
lens  where  its  image  would  appear.  Two  points  so  related 
that  the  image  of  one  of  them,  as  P,  falls  at  the  other,  as  P' , 
are  called  conjugate  foci.  See  also  §  295. 

311.  Virtual  FOCUS.  —  Let  the  candle  and  screen  S  that  were 
used  in  the  experiment  of  the  preceding  section  be  placed  between  a 

convex  lens  and  its  principal 
focus,  and  the  result  shown 
in  Fig.  240  will  be  secured. 
The  rays  diverging  from  the 
point  P  will  not  be  brought 
to  a  focus  by  the  lens,  but 
the  divergence  will  be  greatly 
reduced.  Of  'course  no  im- 
age of  P  can  be  produced 
upon  a  screen;  but  upon  looking  through  the  lens  the  eye  E  locates 
(apparently)  the  image  of  P  at  the  point  P'.  If  the  screen  be  removed 
a  virtual  image  of  the  caudle  ma^  be  seen. 


FIG.  240.  —  The  Conjugate  Focus  of  P  is 
Virtual  and  at  Point  P'. 


LIGHT:   REFLECTION   AND   REFRACTION 


305 


In  the  location  of  an  image  the  eye  is  always  governed 
by  the  divergence  of  rays  which  enter  it  (§  276).  The 
divergence  is  in  this  case  as  if  the  rays  had  come  from  P' 


FIG.  241.  —  A  Concave  Lens  always  Forms  a  Virtual  Focus. 

instead  of  P.  Since  the  point  P'  is  only  the  apparent 
meeting  place  of  the  rays  that  enter  the  eye,  this  point  is 
called  the  virtual  focus  of  P.  If  the  experiment  be 
repeated  with  a  diverging  (concave)  lens,  as  shown  in 
Fig.  241^  the  conjugate  focus  of  P  is  virtual  no  matter 
what  the  position  of  P  may  be,  because  the  effect  of  such 
a  lens  is  always  to  scatter  the  transmitted  light. 

312.  Images  Formed  by  Convex  Lenses.  —  Every  one 
who  has  viewed  the  pictures  projected  by  a  stereopticon 
or  a  moving-picture  machine  has  seen  the  real  images  that 
convex  lenses  are  able  to  produce.  In  the  process  of 
forming  images  the  convex  lens  presents  precisely  as 


FIG.  242.  —  Focusing  a  Real  Image  on  a  Screen  by  Means 

of  a  Convex  Lens. 
21 


306         A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

many  cases  as  the  concave  mirror  (§  295).     These  cases 
are  easily  illustrated  by  experiments. 

1.  Let  a  lighted  candle  C,  Fig.  242,  be  placed  at  a  little  more  than 
twice  the  focal  length  from  a  convex  lens  L.     By  moving  the  screen  5 
back  and  forth  an  image  will  be  found  distinctly  focused  upon  it. 
The  distance  from  the  lens  to  the  image  should  be  measured  and 
compared  with  the  focal  distance. 

When  the  object  is  situated  at  more  than  twice  the  focal 
length  from  a  convex  lens,  the  image  is  at  less  than  twice  and 
more  than  once  the  focal  length  from  the  lens  on  the  opposite 
side,  is  real,  inverted,  and  smaller  than  the  object. 

2.  Let  the  candle  be  placed  at  less  than  twice  but  more  than  once 
the  focal  length  from  the  convex  lens.     The  image  may  be  found  by 
moving  the  screen  away  from  the  lens.     The  distance  to  the  image 
should  be  measured  and  its  position  and  size  noted. 

When  the  object  is  situated  at  more  than  once  and  less 
than  twice  the  focal  length  from  a  convex  lens,  the  image  is 
at  more  than  twice  the  focal  length  on  the  other  side,  is  real, 
inverted,  and  larger  than  the  object. 

3.  Let  the  candle  be  placed  at  less  than  the  focal  length  from  the 
convex  lens.     Of  course  no  image  can  be  produced  upon  the  screen 
since  the  conjugate  foci  of  all  points  in  the  object  are  virtual.     (See 
§  311.)     But  by  allowing  some  of  the  transmitted  light  to  enter  the 
eye  (i.e.  by  looking  through  the  lens),  an  apparent  magnified  image 
may  be  seen  behind  the  lens. 

When  the  object  is  situated  at  a  point  between  a  convex 
lens  and  its  principal  focus,  the  image  is  apparently  behind 
the  lens,  is  virtual,  erect,  and  larger  than  the  object. 

4.  Focus  upon  a  screen  the  images  of  objects  which  are  situated 
at  a  great  distance,  —  clouds,  trees,  buildings,  etc.     If,  now,  the  dis- 
tance from  the  lens  to  the  images  be  measured,  it  will  be  found  prac- 
tically equal  to  the  focal  length  of  the  lens.     Repeat  the  experiment 
with  different  lenses.     Let  the  size  and  position  of  these  images  be 
noted. 


LIGHT:  REFLECTION   AND   REFRACTION        307 

When  an  object  is  at  a  great  distance  from  a  convex  lens, 
its  image  is  at  the  focal  distance  from  the  lens,  is  real, 
inverted,  and  smaller  than  the  object. 

It  will  be  observed  that  this  follows  from  the  fact  that 
the  rays  which  come  from  a  given  point  on  the  object  to 
the  lens  are  practically  parallel  to  each  other  like  rays  of 
sunlight.  This  case  affords  a  good  method  for  determin- 
ing the  focal  length  of  a  lens. 

Since  rays  of  light  which  diverge  from  the  principal  focus 
of  a  convex  lens  become  parallel  to  the  principal  axis  after 
passing  through,  it  follows  that  there  can  be  no  image  of  a 
small  object  placed  at  the  principal  focus. 

313.  Images  Formed  by  Concave  Lenses.  —  It  is  obvious 
from  §  311  that  a  concave  lens  cannot  produce  a  real 
image,  since  it  always  tends  to  scatter  the  light.     This 
fact,  however,  is  of  value  in  the  construction  of  certain 
optical  instruments,  as  will  appear  later. 

Hold  a  concave  lens  between  the  eye  and  a  candle  flame.  No  mat- 
ter how  far  the  flame  is  from  the  lens,  the  only  image  produced  is  a 
small  erect  one  behind  the  lens. 

Hence  the  image  produced  "by  a,  concave  lens  is  always  ap- 
parently on  the  same  side  of  the  lens  as  the  object,  is  virtual, 
erect,  and  smaller  than  the  object. 

314.  Construction  of  Images  Formed  by  Lenses.  —  The 
construction  of  the  images  produced  by  lenses  will  serve 
to  bring  out  clearly  the  nature  of  the  image  in  each  of  the 
cases  illustrated  by  experiment  in  §  312. 

The  different  cases  of  lenses  now  to  be  studied  will  be 
found  to  correspond  closely  to  those  of  curved  mirrors 
which  were  treated  in  section  295.  In  each  case  the  com- 
plete  construction  should  be  accurately  made  on  a  scale 
somewhat  larger  than  that  used  in  the  illustrations  as 
here  given. 


308 


A  HIGH   SCHOOL   COURSE   IN   PHYSICS 


Case  I.  —  When  the  object  is  placed  at  more  than  twice  the 
focal  distance  from  a  convex  lens. 

Let  ED,  Fig.  243,  be  a  convex  lens  whose  centers  of  curvature  are 
at  C  and  C'.     Let  AB  represent  an  object  so  placed  that  the  distance 


FIG.  243.  — Method  of  Locating  the  Image  Produced  by  a  Convex  Lens. 

OI  is  more  than  twice  the  focal  length  OC.  Now  if  rays  A  E  and  BD 
be  drawn  parallel  to  the  principal  axis  ///,  the  refracted  rays  will 
pass  through  the  principal  focus  jP,  which  is  at  C'  (§  308).  Again, 
the*-ays  that  pass  through  the  optical  center  of  the  lens  O  enter  the 
lens  and  emerge  from  it  at  points  where  the  surfaces  are  parallel  and  there- 
fore suffer  no  permanent  change  in  direction.  (See  §  306.)  Let  the 
line  AO  be  produced  through  the  lens  until  it  meets  the  line  EFak  a. 
Thus  a  is  the  conjugate  focus  of  A.  Similarly  b  is  found  to  be  the 
conjugate  focus  of  B.  Therefore  ab  is  the  image  of  the 'object  AB. 

Case  II.  — When  the  object  is  at  more  than  once  and  less 
than  twice  the  focal  distance  from  a  convex  lens. 

In  this  case  the  method  of  construction  is  precisely  the  same  as  in 
the  preceding  one;  but  on  account  of  the  fact  that  the  object  has 
been  brought  nearer  the  lens,  the  divergence  of  the  rays  before  refrac- 


FIG.  244.  —  Illustrating  the  Production  of  a  Real  and  Magnified  Image. 

tion  (which  is  represented  by  angle  OAE)  is  greater  than  before,  and 
hence  they  are  brought  to  a  focus  a  at  a  greater  distance  from  the  lens. 


LIGHT:   REFLECTION   AND   REFRACTION        309 

Figure  244  shows  clearly  the  construction.     Does  the  description  given 
in  §  312  apply  to  the  image  that  is  found  by  construction  ? 

It  is  to  be  observed  that  Case  I  changes  to  Case  II  when 
the  object  is  placed  at  twice  the  focal  length  from  the  lens. 
In  this  instance  the  image  and  obje'ct  are  of  equal  size  and 
are  equidistant  from  the  lens. 

Case  III.  —  When  the  object  is  at  less  than  the  focal  dis- 
tance from  a  convex  lens. 

Two  rays  are  drawn  from  each  of  the  points  A  and  B,  Fig.  245, 
precisely  as  in  the  two  preceding  cases.  But  since  the  angle  of 
divergence  EA  0  is  so  large,  the  lens  is  not  able  to  cause  the  rays  to 
converge  to  a  real  focus  ;  i.e.  the  refracted  rays  EF  and  A  0  never  meet 
after  leaving  the  lens.  If,  however,  the  refracted  rays  enter  an  eye,  an 
apparent  image  of  A  will  be  seen  at  a,  which  is  the  intersection  of  the 


FIG.  245.  — The  Production  of  a  Virtual  Image  by  a  Convex  Lens. 

refracted  rays  when  produced  behind  the  lens.  Since  the  effect  is  an 
illusion,  the  image  is  a  virtual  one.  Does  the  description  given  in 
§  312  apply  to  this  image? 

Case  IV.  —  When  an  object  is  viewed  through  a  concave  lens. 

r~i 


FIG.  246.  —  Constructing  the  Image  Produced  by  a  Concave  Lens. 


310         A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

Rays  A  0  and  BO,  Fig.  246,  are  drawn  as  in  the  preceding  cases. 
But  the  parallel  rays  AE  and  BD  are  turned  away  from  the  principal 
axis  CC'  as  if  their  origin  were  at  the  center  of  curvature  C1  (§  309). 
Thus  the  angle  of  divergence  EA  0  is  increased  by  the  lens  to  the 
value  EaO.  Consequently,  when  the  refracted  light  is  received  by  an 
eye,  the  image  which  is  virtual  and  erect  appears  to  be  behind  the 
lens,  but  nearer  and  always  smaller  than  the  object. 

315.  The  Lens  Equation.  —  The  experiments  of  the  preceding 
sections  have  shown  that  the  position  of  an  image  formed  by  a  lens  is 
determined  by  the  focal  length  of  the  lens  and  the  distance  from  the 
lens  to  the  object.  If  p  represents  the  distance  from  the  lens  to  the 
object,  q  the  distance  from  the  lens  to  the  image,  and  f  the  focal 
length,  then  the  following  relation  between  these  distances  will  be 
found  to  exist  : 

H+J-  (3> 

f     p     q 

Referring  to  Fig.  243,  we  observe  that  the  triangles  -407  and  aOH 

are  similar.     Hence  —  =  ~—  -•     If  the  lens  be  thin  and  a  line  be 
an       Un 

drawn  from  E  to  D,  it  may  be  assumed  to  pass  through  the  point  0. 

EO      OF 

Then  the  triangles  EFO  and  aHF  are  similar.    Therefore,  —77  =  7—  • 

an       tir 

Since  A  I  =  EO,  the  first  member  of  these  two  proportions  are  equal. 
Hence,  by  substituting  p  for  01,  q  for  OH,  and  f  for  OF,  we  obtain 


Clearing    of    fractions,    transposing,    and    dividing    by   pqf  gives 

UI+i. 

/    P     q 

Equation  (3)  is  useful  in  determining  the  focal  length  of  a  lens. 
For  example,  let  the  image  of  an  object  which  is  60  cm.  from  a  lens 
be  focused  40  cm.  from  the  lens.  By  substituting  these  values  for  p 
and  q,  we  find  the  value  of  f,  the  focal  length,  to  be  24  cm. 

316.  Relative  Size  of  Object  and  Image.  —  By  referring 
to  Fig.  243  and  the  above  discussion  the  following  propor- 
tion is  found  to  be  true: 

AI   OI 


LIGHT:  REFLECTION   AND   REFRACTION        311 

But  aHis  the  image  of  AL  Therefore,  the  ratio  of  the 
size  of  the  object  to  the  size  of  the  image  is  equal  to  the  ratio 
of  their  respective  distances  from  the  lens.  This  relation 
may  be  easily  verified  by  experiment. 

EXERCISES 

1.  What   kind    of  mirrors   and   lenses    always    produce    virtual 
images  ? 

2.  Under  what  conditions  do  convex  lenses  produce  real  images? 
When  is  the  image  produced  by  a  convex  lens  virtual  ? 

3.  If  one  half  of  a  convex  lens  be  covered  with  an  opaque  card, 
what  will  be  the  effect  upon  the  real  images  produced  by  it  ?     Test 
your  answer  by  experiment. 

4.  How  can  you  test  a  spectacle  lens  to  ascertain  whether  it  is 
convex  or  concave  ? 

5.  By  the  help  of  equation   (3)    compute  the  focal  length  of  a 
lens  when  the  image  of  a  candle  flame  120  cm.  away  is  focused  at  a 
distance  of  60  cm. 

6.  The  focal  length  of  a  lens  is  50  cm.     If  an  object  is  situated 
at  a  distance  of  75  cm.  from  the  lens,  how  far  from  the  lens  will  its 
image  be  focused  ? 

7.  An  object  is  placed  30  cm.  from  a  lens  whose  focal  length  is 
45  cm.     Locate  the  image  by  employing  equation  (3). 

SUGGESTION. —  When  the  minus  sign  precedes  the  result  obtained 
by  using  the  equation,  the  inference  is  that  the  image  is  virtual. 

8.  What  is  the  height  of  a  tree  350  ft.  away  when  its  image  on  a 
screen  10  in.  from  a  convex  lens  is  2  in.  in  height?  Ans.  70  ft. 

5.    OPTICAL  INSTRUMENTS 

317.   The  Simple  Magnifier  or  Reading  Glass.  —  It  is  a 

common  occurrence  to  see  a  botanist  examining  the  details 
of  a  flower  or  a  jeweler  adjusting  the  minute  parts  of  a  watch 
by  the  help  of  a  convex  lens.  Large  convex  lenses  are 
frequently  used  as  an  aid  in  reading,  and  are  therefore 
often  called  reading  glasses.  In  these  cases  the  object  to 
be  examined  is  placed  a  little  nearer  the  lens  than  the 
principal  focus,  while  the  image  is  viewed  by  placing  the 


312          A  HIGH   SCHOOL  COURSE   IN   PHYSICS, 

eye  on  the  opposite  side  of  the  lens,  as  shown  in  Fig.  240. 
The  instrument  owes  its  importance  to  the  fact  that  a 
magnified  image  is  visible  behind  the  lens,  as  shown  in 
Fig.  245. 

318.  The  Photographic  Camera.  —  Two  important  prin- 
ciples are  employed  in   the  photographic  camera :   (1)  a 
convex  lens  produces  a  real  image  of  objects  placed  beyond 
its  principal  focus,  and  (2)  light  has  the  property  of  pro- 
ducing chemical  changes  in  certain  compounds  of  silver. 

The  camera  is  a  light-proof 
box,  or  chamber,  Fig.  247, 
provided  at  the  front  with 
a  convex  lens  L  and  at  the 
back  with  a  ground-glass 
screen  which  can  be  replaced 
by  a  "sensitized"  plate  or 
film  for  receiving  the  image. 
The  image  is  first  focused  on 
the  screen  by  varying  its  dis- 
tance  from  the  lens,  after 

which  the  sensitive  plate  is  introduced  in  a  light-proof 
holder.  The  shutter  of  the  lens  is  now  closed,  and  the 
cover  of  the  plate-holder  removed.  When  all  is  in  readi- 
ness, the  lens  is  uncovered  for  a  sufficient  time  to  enable 
the  transmitted  light  to  produce  the  desired  effect  upon  the 
plate.  The  plate  is  now  covered  and  taken  to  a  dark  room, 
where  a  "developing"  process  brings  out  a  visible  and  per- 
manent image.  The  plate  thus  treated  is  called  a  "  neg- 
ative "  because  of  the  reversal  of  light  and  shade  in  the 
picture  upon  it,  and  may  be  used  in  the  reproduction  of 
any  number  of  positive  photographs  on  prepared  paper. 

319.  The  Eye.  —  Although  the  eye  is  a  very  complicated 
structure,  its  action  depends  on  one  of  the  simplest  cases 
of  refraction  that  we  have  studied.     The  eye  is  essentially 


LIGHT:  REFLECTION    AND   REFRACTION        313 

a  small  camera  at  the  front  of  which  the  cornea  <7,  Fig.  248, 

the  aqueous  humor  A,  and  the  crystalline  lens  0  take  the 

place  of  the  convex  lens  of  that 

instrument.     When  the  eye  is 

directed  toward  an  object,  a 

small,  real,  and  inverted  image 

is    produced    on    the    retina, 

which  is  an  expansion  of  the 

optic  nerve  N  and  covers  the 

inner  surface  of  the  eyeball  at 

the  back.    This  does  not  mean,         FlG-  248- -Sectional  View 

of  the  Eye. 

however,  that  we  see   things 

upside  down.  The  relative  position  which  we  ascribe  to 
objects  is  the  result  of  experience  aided  by  the  sense  of 
touch,  etc.,  and  the  fact  that  images  are  inverted  on  the 
retina  has  little  effect  on  the  ideas  which  the  impression 
gives  us.  It  will  be  observed  that  the  impressions  remain 
the  same  even  when  the  eye  is  tilted  or  inverted. 

The  eye  adjusts  itself  to  objects  near  and  far  by  chang- 
ing the  focal  length  of  the  crystalline  lens.  When  a 
normal  eye  is  completely  relaxed,  the  lens  has  the  proper 
curvature  for  focusing  light  from  distant  objects  (i.e.  par- 
allel rays)  upon  the  retina.  When,  however,  we  wish  to 
view  an  object  near  at  hand,  as  in  reading,  small  muscles 
within  the  eye  cause  the  curvature  of  the  crystalline  lens 
to  increase  until  the  image  is  again  focused  upon  the  sen- 
sitive retina. 

320.  Spectacles  and  Eyeglasses.  — Although  a  normal 
eye  with  complete  relaxation  focuses  parallel  rays  upon 
the  retina,  as  in  Fig.  249,  it  should  also  be  able  to  focus 
with  ease  light  that  comes  from  an  object  at  a  distance  of 
10  inches  or  more.  The  eye  is  defective  when  it  cannot 
accomplish  the  performance  of  these  functions  without 
unnatural  effort. 


314 


A  HIGH   SCHOOL   COURSE   IN   PHYSICS 


If  the  retina  is  too  far  from  the  crystalline  lens,  parallel 
rays  will  not  be  brought  to  a  focus  upon  it,  but  in  front 
of  it.  This  defect  is  called  myopia,  or  near-sightedness. 
(.Z)  Fig.  250  illustrates  this  con- 
dition. It  is  at  once  obvious  that  the 
crystalline  lens  produces  too  great  a 

FIG.  249.—  A  Normal  Eye  convergence  of  the  light.     This  can 
be  corrected  by  using  a  concave  lens 
of  suitable  curvature,  as   shown   in 
(.2)  Fig.  250.     Hence  concave  spec- 
tacles are  used  to  assist  myopic  eyes. 
Again,  the  eyeball  may  be  too  short 
.(2)  from  front  to  back,  in  which  case  the 
focus  of  parallel  rays  will  be  behind 


when  Relaxed. 


(2) 


FiG.250.-TheMy7picEye  the  retina>  aS  sh°wn  in 

and  its  Correction.  It  is  clear  in  this  instance  that  the 
crystalline  lens  does  not  converge  the 
rays  sufficiently  for  distinct  vision. 
This  defect  is  called  hypermetropia,  or 
far-sightedness.  In  order  to  correct 
the  fault,  a 
convex  lens 
of  suitable 

FIG.  251.  —  The  Hyperme- 

tropic  Eye  and  Its  Cor-  Curvature 

rection.  is  needed, 

which  assists  the  crystalline  lens 
to  produce  an  adequate  conver- 
gence of  the  light,  as  shown  in  (2). 

The  most  prevalent  defect  in 

^  r,  T       .      ,,     ,    FIG.   252.  —An   Astigmatic  Eye 

the  eyes  of  young   people   IS   that        Sees  these  Lines  with  Unequal 

known  as  astigmatism.     Astigma-      Distinctness. 
tism  is  due  to  the  fact  that  the  crystalline  lens  is  not  sym- 
metrical about  its  axis.     In  other  words,  a  vertical  section 
through  the  lens  differs  in  form  from  a  horizontal  section. 


LIGHT:  REFLECTION   AND   REFRACTION        315 


Such  an  eye  sees  the  lines  of  Fig.  252  with  unequal  distinct- 
ness. This  defect  is  corrected  by  the  use  of  a  lens  whose 
vertical  and  horizontal  sections  possess  suitable  curvatures 
to  make  up  for  the  deficiencies  in  the  crystalline  lens. 

321.  The  Compound  Microscope.  —  The  compound  micro- 
scope consists  of  a  convex  lens  (9,  Fig.  253,  of  short  (say -^  in.) 
focal  length,  which  is  called 

the  objective,  and  a  larger  con- 
vex lens  E,  called  the  eye- 
piece. When  the  object  to 
be  viewed,  AB,  is  placed  a 
little  beyond  the  principal 
focus  of  0,  a  ieal,  inverted, 
and  magnified  image  is  pro- 
duced at  ab  (§  312).  When 
this  real  image  is  viewed 
through  the  eye-pie^e,  a 
magnified  virtual  image  is' 
seen  at  a'b' . 

322.  The  Astronomical 
Telescope.  —  The    principal 
part  of  a  modern  astronomi- 
cal  telescope    is    the    large 
convex    lens     0,    Fig.    254, 

called  the  object  glass.  This  is  designed  to  collect  a  large 
amount  of  light  in  order  that  the  real  inverted  image  ab 
that  is  formed  may  be  sufficiently  brilliant.  This  image 


FIG.  254.  —The  Astronomical  Telescope. 


316 


A   HIGH    SCHOOL   COURSE   IN   PHYSICS 


falls  close  to  the  eye-piece  E  whose  function  is  precisely 
the  same  as  in  the  compound  microscope  which  is  described 
in  the  preceding  section.  The  inversion  of  the  image  is  of 
little  consequence  in  astronomical  telescopes,  but  for  view- 
ing terrestrial  objects  this  feature  would  be  a  defect. 

323.    The    Opera    Glass,   or    Galileo's   Telescope.  —  The 
honor  of  having  invented  the  original  form  of  the  tele- 


FIG.  255.  —  Illustrating  the  Principle  of  tke  Opera  Glass. 

scope  belongs  to  Galileo,1  who  constructed  the. first  instru- 
ment about  1610.  Two  Galilean  telescopes  arranged  side 
by  side  form  an  opera  glass  or  a  field  glass.  An  object 
glass  <?,  Fig.  255,  converges  the  light  to  form  a  real  image 
at  ab.  Before  the  rays  reach  the  focus,  however,  they  are 
intercepted  by  a  concave  lens  which  gives  them  a  slight 
divergence  as  they  enter  the  eye.  As  a  consequence,  an 
erect  and  magnified  virtual  image  is  seen  at  a'b' . 

324.    The  Projecting  Lantern.  — The  projecting  lantern, 
Fig.  256,  consists  of  a  powerful  source  of  light  A  whose 


FIG.  256.  —  Illustrating  the  Principle  of  the  Projecting  Lantern. 
1  See  portrait  facing  page  70. 


LIGHT:  REFLECTION   AND   REFRACTION        317 

rays  are  concentrated  upon  a  transparent  picture  B  by  the 
convex  condensing  lenses  L.  At  the  front  of  the  instru- 
ment is  placed  the  projecting  lens  P  which  forms  a  real, 
inverted,  and  enlarged  image  'of  B  upon  the  screen  S. 
The  source  of  light  is  usually  an  electric  arc  lamp  or 
a  calcium  light.  The  latter  is  produced  by  directing  an 
exceedingly  hot  flame,  produced  by  burning  a  mixture  of 
hydrogen  and  oxygen,  against  a  piece  of  lime.  When 
raised  to  a  high  temperature,  the  lime  becomes  intensely 
luminous. 

325.  Binocular  Vision.  —  The  Stereoscope.  —  On  account  of 
the  fact  that  the  two  eyes  are  separated  by  a  distance  of  6  or  7  centi- 
meters, the  images  produced  upon  the  two  retinas  are  not  precisely  alike. 
This  has  the  effect  of  giving  to  an  object  the  appearance  oi  solidity  or 
depth.     Advantage  has  been  taken  of  this  fact  in  the  stereoscope,  an 
instrument  which  is  so  constructed  as  to  present  to  each  eye  a  similar 
image  to  that  which  it  would  receive  if  the  object  itself  were  present. 
A  double  photograph  is  first  made  by  means  of  a  camera  having  two 
objectives  separated  by  a  distance  about  equal  to  that  between  the 
two  eyes.     The  two  pictures  thus  taken  differ  just  as  much  as  would 
the  corresponding  images  upon  the    . 

retinas  of  the  eyes.     These  pictures    P 
are  now  mounted  on  the  stereoscope       /?. 
at  A  and  B,  Fig.  257,  so  as  to  be 
viewed  by  the  two  eyes  through  the    f 
half  -lenses  m  and  n.     These  lenses    FIG.  257.  —  Illustrating  the  Principle 
are  so  adjusted  that  the  images  pro-  of  the  Stereoscope, 

duced  upon  the  retinas  are  related  in  position  precisely  as  in  ordinary 
vision.  On  this  account  a  perfectly  natural  blending  of  the  two  im- 
pressions is  brought  about  as  though  the  object  itself  were  in  the  di- 
rection of  C.  The  observer  is  therefore  conscious  of  the  presence  of 
only  one  photograph,  which  gives  the  effect  of  extension  in  a  degree 
that  is  remarkably  true  to  nature. 

326.  The  Kinetoscope The   kinetoscope,   kinematograph,  or 

moving-picture  machine,    is    a   common   object  in  most    cities  and 
villages.      A  life-like   motion    is  given   to   pictures   projected  on   a 
screen  by  means  of  a  series  of  transparent  photographs  taken  as 
follows : 


318          A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

A  camera  is  provided  with  a  shutter  that  opens  and  closes  auto- 
matically about  12  times  a  second.  The  instrument  also  contains  a 
long,  narrow,  sensitive  film  ivhich  moves  along  about  2  centimeters  while 
the  shutter  is  closed,  and  remains  stationary  while  the  shutter  is  open. 
With  this  a  series  of  pictures  is  taken,  each  of  which  differs  slightly 
from  the  preceding,  provided  any  moving  object  is  in  the  field  of  the  camera. 

These  pictures  are  thrown  upon  a  screen  with  a  projection  lantern 
in  precisely  the  same  order  and  with  the*same  rapidity  as  they  were 
taken.  On  account  of  the  fact  that  the  sensation  produced  by  one 
picture  remains  until  the  next  picture  appears,  the  observer  is  uncon- 
scious of  any  interruption  in  the  illumination  of  the  screen  upon 
which  the  pictures  are  produced. 

EXERCISES 

1.  While  changing  the  attention  from  a  distant  object  to  a  near 
one,  does  the  crystalline  lens  flatten  or  thicken? 

2.  A  photographer  finds  that  the  desired  image  of  a  building  more 
than  covers  the  area  of  the  plate  to  be  used.     How  can  the  size  of  the 
image  be  reduced  to  fit  the  plate? 

3.  When  the  spectacle  lens  used  to  correct  a  myopic  eye  is  placed 
in  front  of  a  normal  eye,  is  the  image  of  a  distant  object  behind  or  in 
front  of  the  retina? 

4.  How  can  you  test  your  eyes  for  astigmatism? 

5.  The  crystalline  lens  becomes  less  elastic  with  age.     Account  for 
the  spectacles  with  double  lenses  which  are  frequently  worn. 

6.  The  picture   projected  on  a  screen  by  a  projecting  lantern  is 
found  to  be  too  large.     Which  way  must  the  instrument  be  moved  in 
order  to  reduce  its  size? 

7.  Why  is  it  necessary  to  "  focus  "  a  microscope  or  a  telescope  upon 
the  object  to  be  viewed  ? 

SUMMARY 

1.  Light  is  reflected  from  polished  surfaces  in  such  a 
manner   that   the    angle    of  reflection  equals  the  angle  of 
incidence  (§§  282  and  283). 

2.  Light  is  diffused  from  unpolished  surfaces.     Tt  is 
the  power  of  diffusing  light  that  renders  objects  visible 
from  different  points  of  observation  when  light  falls  upon 
them  (§  284). 


LIGHT:  REFLECTION   AND   REFRACTION        319 

3.  The  image  produced  by  a  plane  mirror  is  as  far  be- 
hind the  mirror  as  the  object  is  in  front  of  it  (§§  285  to  289). 

4.  The  tendency  of  a  concave  mirror  is  to  collect  rays  of 
light.     Thus  parallel  rays  are  reflected  through  a  common 
point  called  the  principal  focus  (§  29(0. 

5.  The  tendency  of  convex  mirrors  is  to  scatter  light. 
Hence  the  principal  focus  is  unreal,  or  virtual  (§  291). 

6.  The  images  formed  by  a  convex  mirror  are  always 
virtual,  erect,  smaller  than  the  object,  and  behind  the 
mirror  (§  292). 

7.  The  images  formed  by  a  concave  mirror  depend  on 
the  position  of  the  object  relative  to  the  center  of  curva- 
ture (§  294). 

8.  Light  is  refracted  or  bent  toward  the  perpendicu- 
lar to  the  surface  where  it  enters  a  medium  —  as  glass  or 
water  —  from  the  air,  -and  away  from  the  perpendicular 
when  it  emerges  from  the  medium  into  the  air.     Refrac- 
tion is  due  to  the  fact  that  the  speed  of  light  is  less  in 
water,  glass,  etc.,  than  in  air  or  a  vacuum  (§§  297  to  299). 

9.  The  index  of  refraction  of  a  medium  expresses  the 
ratio  of  the  speed  of  light  in   air  to  its  speed  in  that 
medium  (§§  300  to  302). 

10.  Total    reflection    always    takes    place    when  light 
travels  in  one  medium  toward  another  in  which  its  speed 
is   greater,    provided   the   angle    of   incidence    upon   the 
boundary  surface  is  greater  than  the  critical  angle  (§§  303 
to  305). 

11.  Lenses  are  classed  as  convex  or  concave  according  to 
form,  and  as  converging  or  diverging  according  to  their 
effect  on  light  (§  307). 

12.  Convex,  or  converging,  lenses  tend  to  collect  light. 
Rays  which  are  parallel  before  reflection  are  caused  to 


320         A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

pass  through  a  common  point,  called  the  principal  focus 
(§  308). 

13.  Concave,  or  diverging,  lenses  tend  to  scatter  light, 
and  therefore  the  principal  focus  is  unreal  or  virtual  (§  309). 

14.  The  images  formed  by  convex  lenses  depend  upon 
the  relative  position  of  the  object  and  the  principal  focus 
(§  312). 

15.  The  images  formed  by  a  concave  lens  are  always 
virtual,  erect,  smaller  than  the  object,  and  on  the  same  side 
of  the  lens  as  the  object  (§  313). 

16.  Conjugate  focal  distances  p  and  q  are  related  mathe- 
matically to  the  focal  distance  /  as  shown  by  the  following 
equation  : 


17.  The  sizes  of  image  and  object  are  in  proportion  to 
their  respective  distances  from  the  lens  (§  316). 

18.  The  convex  lens  is  employed  in  the  reading  glass, 
camera,   spectacles,  and  the  eye.     Two  or  more  convex 
lenses  are  used  in  the  compound  microscope,    telescope, 
projecting  lantern,  etc.     Concave  lenses  are  used  in  some 
spectacles,  and  in  the  eye-piece  of  the  opera  glass  (§§  317 
to  326). 


CHAPTER   XV 


LIGHT:  COLOR  AND  SPECTRA        ^/ 

1.    DISPERSION   OF   LIGHT:    COLOR 

327.  Decomposition  of  White  Light.  — Let  sunlight  pass 
through  a  narrow  slit  into  a  well-darkened  room  and  fall  obliquely  on 
a  glass  prism,  as  shown 
in  Fig.  258.  As  previ-  \Suniight, 
ously  shown  (§  306),  the 
beam  will  be  refracted; 
but  when  the  refracted 
light  is  allowed  to  fall  on 
a  white  screen,  a  beautiful 
band  of  different  colors  - 
will  be  seen.  Among 
these  will  be  recognized 
red,  orange,  yellow,  green, 


FIG.  258.  —  The  Separation  of  White  Light  into 
its  Components. 


blue,  indigo,  and  violet,  although  there  is  no  sharp  line  of  demarca- 
tion between  them. 

From  this  experiment  we  can  only  infer  that  white  light 
is  a  mixture  of  the  several  colors  seen  on  the  screen.  In 
fact,  if  a  similar  prism  is  available,  the  colors  may  be 

turned  together  again  to  form 
white  light  by  placing  the  two 
prisms  as  shown  in  Fig.  259. 
The  band  of  color  produced 
by  the  separation  of  light  of 
any  kind  is  called  a  spectrum, 
and  the  process  of  separation 

in  which  the  light  of  different  colors  is  refracted  in  vary- 
ing degree  is  called  dispersion. 

22  321 


FIG.  259.  —  Colors  of  the  Spectrum 
Added  to  Produce  White. 


322         A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

328.  Cause  of  Dispersion.  —  Direct  measurements  of  the 
waves  of  light  of  different  colors  show  that  they  are  of  very 
different  lengths,  those  of  red  light  being  longest  and  of 
violet  shortest.     It  is  therefore  clear  that  waves  of  differ- 
ent lengths  are  refracted  unequally  by  the  prism, —  the  red 
or  longest  waves  being  bent  least,  and  the  violet  or  shortest 
waves  most.     The  following  table  shows  the  approximate 
wave  lengths  of  the  various  colors: 

WAVE  LENGTHS  OF  LIGHT 

Red  .  .  .  0.000068  cm.  Green .  .  .  0.000052  cm. 
Orange  .  .  0.000065  cm.  Blue  .  .  .  0.000046  cm. 
Yellow  .  .  0.000058  cm.  Violet .  .  .  0.000040  cm. 

329.  Achromatic  Lenses.  —  When  white  light  passes  through  a 
simple  lens,  it  is  dispersed  as  well  as  refracted,  i.e.  the  violet  light  is 
brought  to  a  focus  somewhat  nearer  the  lens  than  the  red,  since  it  suffers 
the  greater  refraction  (§  328).     On  this  account  the  images  formed  by 
simple  lenses  are  always  fringed  with  color,  which  is  a  serious  fault  in 

optical  instruments.     It  is,  however,  possible  to 
remedy  this  defect  by  combining  two  lenses, — 
one  being  a  double  convex  lens  of  crown  glass,  the 
other  a  plano-concave  lens  (plane  on   one  side, 
FIG.    260.  — Achro-  '  concave  on  the  other)  of  flint  glass,  as  shown  in 
matic  Lens.  Fig.  260.     In  this  lens  system  the  dispersion  pro- 

duced in  one  part  is  just  neutralized  by  the  other, 
while  the  refraction  is  reduced  only  about  one  half.  Such  a  system 
of  lenses  is  called  an  achromatic  lens,  since  all  color  is  eliminated. 

330.  Color  of  Objects.  —  Let  the  spectrum  of  sunlight  be  pro- 
jected on  a  white  screen.    Hold  small  pieces  of  paper  or  cloth  of  differ- 
ent colors  in  the  various  parts  of  the  spectrum,  and  observe  the  effect. 
A  red  object  will  be  a  brilliant  red  when  held  in  the  red  of  the  spectrum, 
and  black  in  other  parts.     Blue  and  violet  objects  show  a  coloration 
only  when  placed  near  the  violet  end  of  the  spectrum.    A  black  object 
will  appear  black  everywhere,  and  a  white  one  reflects  that  color  of 
the  spectrum  in  which  it  is  placed. 

The   experiment   shows   that   the   color   of    an   object 
depends  (1)  on  the  light  which  falls  upon  it  and  (2)  on 


LIGHT:  COLOR  AND  SPECTRA  323 

the  light  it  reflects  to  the  eye.  A  red  object  is  red  be- 
cause it  reflects  mainly  red  light,  all  other  incident  light 
being  absorbed  by  the  material,  i.e.  transformed  into  heat 
(§  259).  If,  therefore,  no  red  light  falls  upon  it,  it  can 
reflect  no  light  and  is  black.  A  black  object  appears  black 
because  it  absorbs  practically  all  colors  alike ;  and  a  white 
body  is  white  because  it  reflects  all  colors  to  the  same 
extent. 

331.  Color  of  Transparent  Bodies.  —Project  the  spectrum  of 
sunlight,  and  hold  pieces  of  glass  of  different  colors  in  the  beam  at 
any  point.     It  will  be  observed  that  a  red  glass,  for  example,  transmits 
mainly  red  light  and  absorbs  the  rest,  that  green   glass  transmits 
mainly  green  light,  etc.     Placing  two  pieces  at  once  in  the  path  of  the 
light  results  in  the  transmission  of  that  light  only  which  neither  can 
absorb. 

It  is  therefore  clear  that  the  color  of  a  transparent  body 
depends  on  the  color  of  the  light  which  it  transmits.  The 
color  that  we  see  in  any  body  is  the  combination  of  all  the 
light  that  is  not  absorbed  by  that  body.  A  colorless  body  is 
one  that  transmits  all  colors  equally. 

332.  Complementary  Colors.  —  Project  the  spectrum  of  sunlight 
or  the  light  from  an  electric  arc, 

and  reunite  the  colors  by  means  of 
a  similar  prism,  as  shown  in  Fig. 
261.  The  result  is,  of  course, 
white.  Now  place  a  card  C  be- 
tween the  prisms,  and  let  it  in- 

,-,,.'  .         _,         FIG.  261.— Production  of  Comple- 

tercept  the  red  light   only.     The  mentary  Colors. 

spot  of  light  on  the  screen  S  turns 

to  bluish  green.    If  the  card  is  now  caused  to  cut  out  the  violet  and 

blue,  a  greenish  yellow  results. 

Two  colors,  as  red  and  bluish  green,  into  which  white 
light  can  be  resolved  are  called  complementary  colors.  The 
union  of  complementary  colors  results  in  the  production  of 
white.  When  any  color,  or  combination'  of  colors,  is 


324          A  HIGH   SCHOOL  COURSE   IN    PHYSICS 

removed  from  the  spectrum  of  white  light,  the  color  pro- 
duced by  combining  the  remaining  colors  is  the  comple- 
mentary of  the  part  removed.  The  two  complementary 
colors  most  easily  obtainable  are  yellow  and  blue.  If  one 
half  of  a  circular  disk  be  colored  blue  and  the  other  half 
yellow,  the  color  effects  may  be  combined  by  rotating  the 
disk  on  a  whirling  machine.  If  the  whirling  disk  be 
strongly  illuminated,  it  appears  white. 

TABLE  OF  COMPLEMENTARY  COLORS 

Red  and  bluish  green.  Green  and  purple. 

Orange  and  greenish  blue.  Violet  and  greenish  yellow. 

Yellow  and  blue. 

333.  Color  of  Pigments.  —  The  color  of  pigments  used 
in  paints  and  coloring  materials  is  due  to  their  power  of 
absorbing  incident  light,  similar  to  that  shown  in  §  330  in 
the  case  of  colored  cloth  and  paper.     The  mixing  of  pig- 
ments is  a  very  different  thing  from  mixing   lights   or 
colors.     A  striking  example  is  furnished  by  pigments  that 
reflect  the  complementary  colors  blue  and  yellow. 

Pulverize  pieces  of  yellow  and  blue  crayon,  but  keep  the 
powders  separate.  If  now  about  equal  portions  of  the  two 
powders  be  thoroughly  mixed  together,  a  bright  green 
appears. 

The  cause  of  the  phenomenon  is  the  imperfect  absorption 
of  the  pigments.  The  yellow  powder  subtracts  from  white 
light  all  except  yellow  and  green,  and  the  blue  powder  sub- 
tracts all  except  blue  and  green.  Hence  the  only  color  not 
absorbed  by  one  powder  or  the  other  is  green. 

2.    SPECTRA 

334.  The  Solar  Spectrum.  —  The  spectrum  of  sunlight, 
or  the  solar  spectrum,  frequently  presents  itself  in  nature 
in  the  rainbow.     In  the  production  of  the  rainbow  the  sun- 


LIGHT:  COLOR  AND   SPECTRA 


325 


light  is  dispersed  by  spherical  raindrops.  Light  from  the 
sun  .strikes  a  drop  at  A,  Fig.  262,  and  is  refracted  to  B. 
At  B  a  portion  of  the 
light  is  reflected  to  (7, 
where  it  emerges  and 
enters  the  eye  E.  At 
the  points  A  and  0  dis- 
persion accompanies  re- 
fraction, and  the  red 
light  takes  the  direction 
rrr,  while  the  violet  fol- 
lows the  path  vvv.  A 
study  of  Fig.  263  will 
show  that  the  eye  re- 
ceives the  red  from  a  greater  angle  of  elevation  than  the 
violet ;  hence  red  lies  at  the  outside  of  the  rainbow. 


FIG.  262.  —  Dispersion  of  Sunlight  by 
a  Raindrop. 


FIG.  263.  —  Showing  the  Spectrum  Colors  in  their  Relative  Positions  in  the 

Rainbow. 

The  secondary  rainbow  often  accompanies  the  primary 
one.     In  this  case,  as  Fig.  264  shows,  light  suffers  two 


326 


A  HIGH   SCHOOL   COURSE   IN   PHYSICS 


reflections  within  the  drop  and  emerges  with  the  violet  at 
the  greater  angle  of  elevation ;  hence  here  in  the  secondary 
bow  the  red  band  is  the  inner  circle  of  color. 


FIG.  264.  —  Showing  the  Formation  of  a  Secondary  Bow. 

335.  Absorption  of  Light  by  the  Medium.  —  Let  a  narrow, 
horizontal  beam  of  sunlight  come  into  a  darkened  room  through  a 
vertical  slit  S,  Fig.  265,  and  pass  through  a  convex  lens  L  to  the 
carbon  disulphide  prism  P.  (The  dispersion  produced  by  a  glass 


Sunlight 


FIG.  265.  —  Method  of  Projecting  the  Solar  Spectrum 

prism  is  too  small  to  give  satisfactory  results.)  Reflect  some  of  the 
light  from  the  back  of  the  prism  to  the  screen  AB  where  the  spec- 
trum is  to  be  formed,  and  adjust  the  lens  until  it  focuses  a  sharp 


LIGHT:  COLOR  AND   SPECTRA  327 

image  of  the  slit  on  the  screen.     A  good  spectrum  can  now  be  produced 
by  setting  the  prism  in  the  position  shown  in  the  figure. 

While  the  spectrum  is  on  the  screen,  place  before  the  slit  at  C  a 
flat  tank  or  bottle  containing  a  dilute  solution  of  chlorophyl,  made  by 
soaking  a  quantity  of  grass  in  warm  alcohol.  It  will  be  observed 
that  a  broad  dark  band  is  formed  in  the  red  portion  of  the  spectrum 
and  a  less  marked  band  appears  in  the  green. 

It  appears  from  this  experiment  that  a  certain  portion 
of  the  solar  spectrum  may  be  removed  by  the  absorption 
of  certain  wave  lengths  in  the  medium  through  which  the 
light  passes.  The  resulting  spectrum  is  called  an  absorp- 
tion spectrum  to  distinguish  it  from  the  continuous  spec- 
trum that  is  obtained  when  none  of  the  light  is  absorbed. 
Continuous  spectra  are,  in  general,  given  off  by  all  lu- 
minous solids  and  liquids.  It  will  appear  presently  that 
the  solar  spectrum  is  an  absorption  spectrum. 

336.  The  Solar  Spectrum  in  Detail.  —  The  most  interest- 
ing and  remarkable  features  of  the  solar  spectrum  are  not 
revealed  in  the  rainbow,  nor  in  the  spectrum  ordinarily 
projected  by  a  prism,  on  account  of  the  overlapping  color 
bands.  In  order  to  produce  a  pure  spectrum  in  which  the 
colors  are  more  distinctly  separated  than  before,  it  is  nec- 
essary to  work  with  a  narrow  slit  which  is  very  sharply 
focused  by  a  lens  on  a  white  screen. 

Let  the  solar  spectrum  be  projected  as  before,  but  make  the  slit 
about  one  half  a  millimeter  in  width  and  use  a  long  focus  (about 
50  centimeters)  lens.  Place  the  screen  about  1  meter  from  the  lens  and 
prism.  By  making  all  the  adjustments  with  care,  it  will  easily  be 
seen  that  the  spectrum  is  crossed  vertically  by  many  dark  lines.  By 
moving  the  screen  back  and  forth  the  best  position  for  showing  the 
lines  can  readily  be  found. 

These  dark  lines  across  the  solar  spectrum  are  known  as 
Fraunhofer  lines  in  honor  of  the  German  astronomer 
who  first  mapped  them  out  about  1814.  When  the  spec- 
trum is  magnified  sufficiently,  hundreds  of  these  lines 


328          A  HIGH   SCHOOL   COURSE   IN   PHYSICS 

become  visible.  The  presence  of  the  Fraunhofer  lines 
shows  that  certain  wave  lengths  are  either  absent 
entirely  from  sunlight  or  that  they  are  go  weak  as  to 
appear  dark  by  contrast.  The  solar  spectrum  is  thus 
observed  to  be  an  absorption  spectrum. 

337.  Fraunhofer  Lines  Produced  by  Absorption. — The 
explanation  of  the  dark  lines  of  the  solar  spectrum  is  due 
to  Stokes  and  Kirchhoff,  the  former  an  English,  the  latter 
a  German,  physicist.  The  theory  is  based  on  the  follow- 
ing experiment: 

Project  a  well-defined  solar  spectrum  as  in  the  preceding  experi- 
ment. Now  sprinkle  common  salt  (sodium  chloride)  on  the  wick 
of  an  alcohol  lamp,  place  the  lamp  just  below  the  slit  at  C,  Fig.  265, 
and  ignite  the  alcohol.  A  bright  yellow  light  will  be  emitted  by 
the  volatilized  salt,  but  on  the  screen  will  appear  a  darkening  of  a 
Fraunhofer  line  in  the  yellow  part  of  the  spectrum. 

It  thus  appears  that  when  light  passes  through  the  so- 
called  sodium  flame,  the  absorption  of  a  certain  wave 
length  takes  place.  In  the  same  manner  the  light  from 
the  white-hot  central  mass  of  the  sun,  which  alone  would 
give  a  spectrum  without  dark  lines,  passes  through  sur- 
rounding vapors,  each  of  which,  like  the  sodium  vapor, 
has  the  power  of  removing  light  of  certain  wave  lengths. 
The  wave  lengths  absorbed  by  a  heated  gas  or  vapor  are  pre- 
cisely those  which  the 
vapor  itself  is  capable  of 
emitting. 

338.  Bright-line  Spec- 
tra. —  Prepare    an    alcohol 
lamp  for  producing  yellow 
light  as    in    the    preceding 
FIG.  266.  —  Apparatus  Arranged  for  Bright-      experiment.     Place  the  lens 
line  Spectra.  L>  Fig>  266,  so  that  the  slit 

A  is  at  its  principal  focus,  and  set  the  prism  as  shown  in  the  figure. 
Focus  a  small  telescope  T  on  a  distant  object,  and  then  place  it  for 


LIGHT:  COLOR  AND  SPECTRA  329 

receiving  light  from  the  prism.  Place  the  lamp  at  F,  and  ignite 
the  alcohol.  If  all  is  properly  adjusted,  a  bright  double  yellow  line 
will  be  visible  in  the  telescope.  This  is  the  spectrum  of  incandes- 
cent sodium  vapor.  Repeat  the  experiment  by  using  strontium 
chloride  in  another  lamp.  Red  and  blue  lines  should  appear.  By 
using  calcium  chloride,  red  and  green  lines  become  visible.  The 
yellow  sodium  spectrum  will  always  appear,  since  sodium  exists  as  an 
impurity  in  the  lamp  wick  and  in  practically  all  salts. 

These  experiments  show  that  incandescent  vapors  emit 
light  in  which  only  certain  wave  lengths  are  present.  The 
spectra  of  such  bodies  are  therefore  bright-line  spectra. 
Since  these  spectra  are  characteristic  of  the  chemical  ele- 
ments, many  substances  can  be  identified  by  the  wave 
lengths  of  the  light  which  their  incandescent  vapors  emit. 
The  spectra  of  the  chemical  elements  are  as  well  known  as 
their  densities,  specific  heats,  and  other  physical  and  chemi- 
cal properties.  Several  new  elements  have  been  discovered 
by  the  observation  of  bright  spectral  lines  which  could  not 
have  been  produced  by  any  known  substances.  An  incan- 
descent vapor  also  affords  a  convenient  means  of  obtaining 
light  of  one  wave  length  or  color,  i.e.  monochromatic  light. 

339.  Solar  Elements.  —  Since  the  spectra  of  many  incan- 
descent vapors  have  been  examined  and  compared  with 
the  Fraunhofer  lines  of  the  solar  spectrum,  most  of  these 
lines  have  been  accounted  for  just  as  the  existence  of 
incandescent  sodium  vapor  surrounding  the  sun  accounts 
for  the  dark  sodium  line.  (See  §  337.)  If  the  corre- 
spondence between  lines  of  the  solar  spectrum,  shown  in 
the  middle  in  Fig.  267,  and  the  spectrum  of  iron  vapor,  be 
noted,  there  can  be  no  doubt  as  to  the  existence  of  iron 


FIG.  267.  —  Showing  the  Coincidence  of  the  Bright  Lines  of  the  Spectrum  of 
k  Iron  with  Some  of  the  Dark  Lines  of  the  Solar  Spectrum. 


330          A   HIGH   SCHOOL   COURSE   IN   PHYSICS 

in  the  sun.  Other  solar  elements  are  calcium,  hydrogen, 
sodium,  nickel,  magnesium,  cobalt,  silicon,  aluminium, 
titanium,  chromium,  manganese,  carbon,  barium,  silver, 
zinc,  and  many  others.  It  is  an  interesting  fact  that  the 
element  helium  was  first  discovered  in  the  sun  by  Sir 
Norman  Lockyer,  of  England,  by  means  of  its  dark  lines 
in  the  solar  spectrum.  Later,  in  1895,  Sir  William 
Ramsay  discovered  an  element  that  gave  the  same  spec- 
trum, and  which  was  therefore  given  the  same  name. 
Helium  is  at  present  a  comparatively  well-known  gas  that 
can  be  produced  in  all  chemical  laboratories. 

The  instrument  by  which  the  spectra  of  celestial  and 
terrestrial  objects  are  produced  is  called  the  spectroscope. 
By  combining  the  spectroscope  with  the  telescope,  astrono- 
mers have  succeeded  in  ascertaining  to  some  extent  the 
composition  of  many  stars  and  in  collecting  other  informa- 
tion of  great  scientific  value. 

EXERCISES 

1.  What  kind  of  a  spectrum  should  moonlight  give  ? 

2.  What  kind  of  a  spectrum  would   you  expect  to  obtain  by  dis- 
persing the  light  from  a  live  coal  by  means  of  a  prism? 

3.  The  luminosity  of  an  oil  or  gas  flame  is  due  to  heated  particles 
of  carbon.     What  should  be  the  nature  of  the  spectrum  of  a  kerosene 
lamp  flame  ? 

4.  What  is  the  position  of  the  sun  relative  to  falling  rain  and  an 
observer  when  a  rainbow  is  seen? 

5.  Why  do  colored  fabrics  often  appear  different  when  viewed  by 
artificial  light? 

6.  If  the  waves  producing  the  sensation  of  red  were  all  absorbed 
from  sunlight,  what  color  would  remain?     What  color  would  objects 
that  were  formerly  red  appear  to  have  ? 

3.    INTERFERENCE    OF  LIGHT 

340.  Color  Bands  by  Interference.  —  We  have  seen  how 
composite  light  can  be  separated  into  its  component  colors 


LIGHT:  COLOR  AND   SPECTRA 


331 


by  means  of  a  prism,  but  there 
still  remain  a  great  many  cases 
of  color  formation  which  are  not 
at  all  due  to  dispersion  or  ab- 
sorption. 

Let  two  strips  of  plate  glass,  A  and 
B,  Fig.  268,  about  1  inch  wide  and  5 
inches  long,  be  separated  at  one  end  by 
a  piece  of  tissue  paper  C  (exaggerated 
in  cut)  and  the  other  ends  clamped 
tightly  together.  (The  plate  glass 
covers  accompanying  sets  of  small 
weights  may  be  used.)  Produce  a  yel- 
low sodium  flame  by  bringing  in  con- 
tact with  a  Bunsen  or  alcohol  flame  a 


FIG.  268.  —  Alternate  Dark  and 
Bright  Bands  Produced  by  In- 
terference of  Light  Waves. 

piece  of  asbestos  D,  wet  with  a  solution  of  common  salt.     Now  hold 

the  glass  strips  behind  the  flame,  and  observe  the  images  produced. 

A  series  of  dark  and  yellow  bands  will  be  seen  extending  transversely 

across  the  glass. 

The  explanation  of 
this  interesting  phe- 
nomenon depends  on 
the  wave  theory  of 
light.  The  flame,  as 
we  know,  sends  out 
only  yellow  light 
(§338).  This  is  in 
part  transmitted  and 
in  part  reflected  at 
each  glass  surface 
(§282).  Let  AB  and 
AC,  Fig.  269,  repre- 
sent the  interior  sur- 
faces of  the  plates  much 
enlarged,  and  let  the 
wave  line  ab  represent 


Interference. 


Re-enforcement, 


interference. 


Re-enforcement 


FIG.  269.  —  Diagram  Showing  Points  of  In- 
terference  and  Reinforcement. 


332          A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

the  light  reflected  at  the  point  a  on  the  surface  AC. 
Also  let  the  dotted  line  a'b  represent  the  waves  reflected 
at  a'  on  the  surface  AB.  These  two  trains  of  waves  ob- 
viously interfere  and  destroy  each  other  just  as  do  two 
trains  of  sound  waves  when  they  coincide  in  this  manner 
(§  190).  Therefore,  since  at  this  place  on  the  glass 
plates  the  light  waves  cancel  each  other,  we  see  a  dark 
band.  But  at  the  point  c  the  case  is  different.  Since 
the  thickness  of  the  air  film  is  here  a  quarter  of  a  wave 
length  greater  than  at  a,  the  train  of  waves  reflected 
from  c'  coincides  with  that  reflected  at  c  in  such  a  manner 
as  to  produce  reenforcement.  Hence  at  this  point  we  see 
the  bright  yellow  band.  Similarly  the  wave  trains  re- 
flected at  e  and  e'  interfere  and  cancel,  while  at  g  and  g1 
we  again  find  reenforcement.  Thus  the  dark  and  bright 
bands  appear  alternately  in  the  plates. 

341.  Color  Bands  in  Soap  Films. — The  following  ex- 
periment can  easily  be  performed  by  letting  a  soap  film  take 
the  place  of  the  wedge-shaped  air  film  of  the  preceding 
experiment. 

Let  a  film  of  soapy  water  be  produced  on  a  wire  loop  as  described 
in  §  130.  Hold  the  film  behind  a  yellow  sodium  flame,  and  observe 
the  image  in  the  film  by  reflected  light.  Many  narrow  bands  of 
yellow  will  appear  which  continue  to  grow  broader  and  farther  apart 
until  the  film  breaks. 

When  the  wire  loop  is  held  in  a  vertical  position,  the 
liquid  runs  slowly  down  from  the  top,  forming  a  wedge- 
shaped  film  that  is  thinnest  at  the  upper  edge.  One  por- 
tion of  the  yellow  incident  light  is  reflected  from  the 
front  surface,  and  another  portion  from  the  back,  after 
traversing  the  thickness  of  the  film.  Hence  the  portion 
reflected  from  the  back  passes  through  the  film  twice. 
Now  if  the  two  reflected  trains  interfere,  a  dark  band  is 


LIGHT:  COLOR  AND  SPECTRA  333 

produced ;  but  if  they  reenforce,  a  bright  band  of  yellow 
appears,  precisely  as  described  in  the  preceding  section. 

342.  White  Light  Decomposed  by  Interference.  —  The 
result  of  the  experiment  described  in  §  341  leads  to  the 
explanation  of  the  beautiful  coloration  produced  when 
the  sunlight  falls  on  soap  bubbles,  oil  films  on  water,  etc. 

Let  sunlight  fall  upon  the  plates  of  glass  arranged  as  in  §340. 
Variegated  bands  occurring  alternately  will  appear  instead  of  the 
yellow  bands  obtained  when  sodium  light  is  used. 

On  account  of  the  fact  that  sunlight,  or  white  light,  is 
a  composite  of  different  wave  lengths,  the  interference 
of  the  reflected  waves  of  red,  for  example,  takes  place  at 
a  different  point  from  that  of  the  yellow.  When  the 
waves  of  red  are  canceled  by  interference,  the  comple- 
mentary color,  or  bluish  green,  is  left.  At  the  point  where 
waves  of  yellow  interfere,  its  complement,  or  blue,  appears. 
Thus  bands  consisting  of  the  complements  of  all  the 
spectral  colors  are  produced. 

SUMMARY 

V.  White  light  is  decomposed  into  the  spectral  colors  by 
passing  through  a  triangular  prism  of  glass,  water,  ice,  etc. 
The  cause  of  dispersion  lies  in  the  fact  that  different  wave 
lengths  are  retarded  different  amounts,  and  hence  are 
refracted  in  differing  degree  by  the  prism.  Long  waves 
(red)  suffer  the  greatest  refraction,  while  the  shorter 
waves  (violet)  suffer  least  (§§  327  to  329). 

2.  The  color  of  an  object  depends  both  on  the  light 
which  falls  upon  it  and  that  which  it  reflects  to  the  eye. 
Various  wave  lengths  are  absorbed  by  all  except  white 
bodies  (§  330). 

3.  The  color  of  a  transparent  body  depends  on  the  color 
of  the  light  which  it  transmits  (§  331). 


334          A  HIGH   SCHOOL   COURSE   IN   PHYSICS 

4.  Complementary  colors  are  those  which  when  added 
in  proper  proportions  produce  white  (§  332). 

5.  The  color  of  pigments  used  in  the  manufacture  of 
paints,  etc.,  is  due  to  the  absorbing  power  of  the  material 
(§  333). 

6.  The  solar  spectrum  is  the  spectrum  of  sunlight.     The 
rainbow  is  an  example.     The  pure  spectrum  is  crossed  by 
numerous  dark  lines  produced  by  the  absorption  of  certain 
wave  lengths  by  heated  gases  in  the  solar  atmosphere. 
These  lines  reveal  the  chemical  elements  in  the  sun's  com- 
position (§§  334  to  339). 

7.  Interference  of  light  is  brought  about  by  the  coinci- 
dence of  waves  in  such  a  manner  as  to  weaken  or  cancel 
each  other.     The  alternate  interference  and  reinforcement 
of  light  waves  in  thin  wedge-shaped  films  gives  rise  to  the 
colors  seen  in  oil  films,  soap  bubbles,  etc.  (§§  340  to  342). 


CHAPTER   XVI 
ELECTROSTATICS 

1.    ELECTRIFICATION   AND    ELECTRICAL   CHARGES 

343.  An  Electric  Charge  Produced.  —  It  is  a  matter  of 
common  observation  that  a  hard  rubber  comb  acquires  the 
power  of  attracting  bits  of  tissue  paper  and  other  very 
light  bodies  simply  by  being  drawn  through  dry  hair.  The 
comb  may  also  be  put  into  this  condition  by  being  rubbed 
with  silk  or  flannel.  Knowledge  of  this  phenomenon 
dates  from  the  Greeks  of  600  B.C.,  when  it  was  known 
that  amber  would  attract  light  objects  after  having  been 
rubbed  with  silk.  No  advance  was  made  in  the  science 
of  electricity  until  the  time  of  Sir  William  Gilbert  of 
England,  about  1600  A.D.  At  this  time  Gilbert  found 
that  a  great  many  substances,  as  glass,  ebonite,  sealing 
wax,  etc.,  could  be  given  this  power  of  attraction  by 
rubbing  with  fur,  wool,  or  silk. 

Test  rods  of  glass,  ebonite,  sealing  wax,  etc.,  for  the  power  of 
attracting  small  pieces  of  dry  pith  before  and  after  being  rubbed 
with  silk,  fur,  and  flannel. 
See  Fig.  270. 

When  a  body  pos- 
sesses the  property  of 
attracting  light  objects, 
as  hair,  pith  balls,  and 
bits  of  paper,  it  is  said 
to  be  electrified.  The  Fia  27(X  ~  Action  of  an  Electrified  Glass  Rod- 
change  is  brought  about  by  a  charge  of  electricity.  The 
process  by  which  a  body  is  electrified  is  called  electrifica- 

335 


336          A   HIGH   SCHOOL   COURSE   IN   PHYSICS 

tion.     These  terms  are  all  derived  from  the  Greek  word 
electron,  meaning  amber. 

344.  Electrical  Charges  of  Two  Kinds.  —  Procure  two  large 

sticks  of  sealing  wax  and 
a  glass  rod.  Electrify  a 
stick  of  sealing  wax  by  rub- 
bing it  with  a  flannel  or 
fur  and  suspend  it  in  a  wire 
stirrup  as  shown  in  Fig. 
^  271.  Rub  the  other  stick 

Jto.  271.  -  Repulsion  between  Two  Simi-       of  sealinS  wax  and  brinS  it; 
larly  Charged  Bodies.  near  the  suspended  one.     A 

decided   repulsion    will    be 

observed.     Now  rub  the  glass  rod  with  silk  and  bring  it  near  the  sus- 
pended wax.     Attraction  takes  place. 

The  experiment  shows  that  the  electrified  glass  and 
sealing  wax  cannot  be  in  precisely  the  same  condition, 
since  they  act  differently  toward  the  suspended  body. 
This  difference  in  the  behavior  of  electrified  bodies  is 
said  to  be  due  to  the  kind  of  charge  developed  when  they 
are  rubbed.  Thus  glass  is  said  to  be  charged  positively 
when  rubbed  with  silk ;  and  sealing  wax,  negatively  when 
rubbed  with  flannel  or  fur.  Some  specimens  of  glass  are 
electrified  negatively  when  rubbed  with  cat's  fur  or  flannel. 

345.  A  Law  of  Electric  Action.  —  The  preceding  experi- 
ment shows  tluit  two  sticks  of   sealing  wax   repel   each 
other  after  being  charged  negatively.     In  the  same  man- 
ner two  positively  charged  glass  rods  will  show  a  repulsion. 
But  any  negative  charge  will  attract  a  positive  charge, 
just  as  the  glass  attracts  the  suspended  sealing  wax  when 
both  are  electrified.     Hence,  we  may  infer  that  electrical 
charges  of  a  similar  kind  repel  each  other,  and  those  that  are 
dissimilar  attract. 

346.  The  Electroscope.  —  In  order  to  detect  the  presence 
of  a  charge  upon  a  body,  an  instrument  called  the  electro- 


ELECTROSTATICS  337 

scope  is  employed.     The  gold-leaf  electroscope  is  shown  in 
Fig.  272.     This  instrument  consists  of  a  metal  rod  which 
penetrates  the  rubber  stopper  of  a  flask.     At  the  top  the 
rod  terminates  in  a  plate  or  ball,  and  at 
the   lower   end   it  is  provided  with  two 
strips  of  gold  foil.     Under   ordinary  con- 
ditions the  leaves  hang  parallel ;  but  when 
an  electrical  charge  is  brought  near,  they 
diverge  and  thus  show   the   presence   of 
electrification. 

347.  Determining  the  Kind  of  Charge.  FIG.  ^72. —  The 
—  The  electroscope  is  frequently  used  to  Gold-leaf  Eiec- 

i.   •       ^      i  •    j      F  troscope. 

ascertain  the  kind  of  charge  on  an  elec- 
trified body.     The  following  experiment  will  show  how 
this  can  be  done. 

Make  a  proof  plane  by  sealing  a  small  coin  or  tin  disk  to  the  end 
of  a  small  glass  rod  to  be' used  as  a  handle.  Touch  the  disk  of  the 
proof  plane  to  a  positively  charged  glass  rod,  and  then  to  the  plate 
of  the  electroscope.  The  divergence  of  the  leaves  will  show  that  the 
instrument  is  charged.  Now,  if  the  charged  glass  rod  is  carefully 
brought  near  the  electroscope,  the  divergence  will  increase ;  but  if 
the  negatively  charged  sealing  wax  is  brought  near,  the  divergence 
at  once  decreases.  While  the  electroscope  is  thus  charged,  bring  up 
an  electrified  rod  of  ebonite  or  a  charged  rubber  comb,  and  note 
whether  the  divergence  increases  or  decreases.  In  this  case  the  charge 
on  the  ebonite  will  be  found  to  act  like  that  on  the  sealing  wax  and 
hence  produce  a  decreased  divergence  of  the  leaves. 

In  a  word,  the  nature  of  an  unknown  charge  is  deter- 
mined by  observing  whether  its  effect  on  a  charged  electro- 
scope is  like  that  of  a  positively  charged  glass  rod  or  the 
negatively  charged  sealing  wax. 

348.  Positive  and  Negative 
Charges  Developed  Simultane- 
OUsly.-Let  a  flannel  cap  about  6 

Time.  inches  long  be  made  that  will  fit  closely 

23 


338         A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

over  the  end  of  an  ebonite  rod  as  in  Fig.  273.  A  represents  a  silk 
thread  by  which  the  cap  may  be  handled.  Now  turn  the  rod  in  the 
cap  to  electrify  it  and,  without  removing  the  cap,  hold  the  rod  near 
the  plate  of  the  electroscope.  Little  or  no  charge  will  be  detected. 
Remove  the  cap  by  means  of  the  thread  A  and  present  it  to  the  elec- 
troscope. When  tested  as  in  §  347,  the  cap  will  be  found  to  be  charged 
positively.  If  the  rod  be  now  tested,  a  negative  charge  will  be  found. 

On  account  of  the  fact  that  the  rod  and  cap  together 
produce  no  effect  on  the  electroscope,  we  may  infer  that 
their  charges  exactly  cancel.  The  conclusion  is,  there- 
fore, that  when  a  certain  quantity  of  one  kind  of  electrifica- 
tion is  produced  by  rubbing  a  rod,  an  equal  amount  of  the 
opposite  kind  appears  on  the  object  with  which  it  is  rubbed. 

349.   Charging     by     Contact.  —  By 

means  of  a  silk  thread  or  silk  fiber  suspend 
a  small  ball  of  dry  elder  pith  as  shown  in 
Fig.  274.  Hold  near  the  ball  a  positively 
charged  glass  rod.  The  ball  is  at  first  at- 
tracted to  the  rod  and  then  violently  repelled. 
The  repulsion  shows  that  the  ball  by  contact 
with  the  rod  has  been  charged  with  a  kind  of 
electrification  similar  to  that  in  the  rod.  In 
FIG.  274.  —  Experiment  this  case  the  charge  is  positive.  Likewise  the 
with  a  Suspended  ball  will  be  charged  negatively  by  contact 
with  a  negatively  charged  body. 

The  experiment  shows  clearly  that  when  an  insulated 
body  comes  in  contact  with  a  charged  body,  it  becomes 
charged  with  electricity  of  the  same  kind  as  that  on  the 
charged  body.  In  this  way  the  proof  plane  used  in  §  347 
carried  from  the  glass  rod  to  the  electroscope  an  electrical 
charge  of  the  same  kind  as  that  on  the  glass. 

350.  Conductors  and  Insulators.  —  Select  a  number  of  pieces 
of  wood,  metal,  glass,  ebonite,  cardboard,  leather,  etc.  Let  one  end  of 
a  metal  rod  be  supported  on  the  electroscope  and  the  other  end  on  an 
ebonite  rod  A,  Fig.  275.  Bring  a  charged  body  R  against  the  end  of 
the  rod  opposite  the  electroscope.  A  sudden  divergence  of  the  leaves 


ELECTROSTATICS  339 

shows  the  transfer  of  a  charge  from  R  to  the  instrument.  Experi- 
ment with  a  rod  of  wood  in  the  same  manner.  The  divergence,  if  any 
is  produced,  takes  place  more  slowly  than  before  if  the  wood  is  dry. 
Glass  and  ebonite  should  be  found 
not  to  transfer  any  appreciable 
charge  unless  their  surfaces  are 
moist. 

Some  substances  have  the 

power  to  transfer  charges  of     FIG.  275.  — Testing  the  Conductivity 

electricity,  while  others  do  of  Rods, 

not.  Metals  are  shown  to  be  good  conductors ;  dry  wood 
is  a  poor  conductor,  while  glass  and  ebonite  are  practically 
non-conductors.  Non-conductors  are  called  insulators. 
Among  the  best  insulators  may  be  mentioned  dry  air, 
glass,  mica,  shellac,  silk,  rubber,  porcelain,  paraffin,  and  oils. 
Substances  of  all  degrees  of  conductivity  exist,  varying 
from  the  best  conductors  down  to  the  best  insulators. 

EXERCISES 

1.  Place  a  piece  of  paper  against  the  wall  and  stroke  it  with  cat's 
fur.^Lt  will  be  found  to  adhere  to  the  wall   for   several  minutes. 
Explain. 

2.  Why  is  it  more  difficult  to  brush  lint  from  clothing  in  cold  dry 
weather  than  at  other  times  ? 

3.  Draw  a  rubber  comb  through  dry  hair  and  test  the  charge  de- 
veloped on  the  comb.     Now  present  a  positively  charged  rod  to  the 
charged  hair  and  observe  whether  it  is  repelled  or  attracted.     Com- 
pare this  experiment  with  that  of  §  348. 

4.  After  walking  on  a  silk  or  woolen  rug  or  over  a  glass  floor,  one 
often  finds  that  sparks  will  jump  between  the  finger  and  a  gas  fixture 
near  which  it  is  held.     The  gas  may  even  be  lighted  in  this  manner. 
Explain. 

5.  Support  a  dry  pane  of  glass  about  1  in.  above  a  quantity  of 
small  pieces  of  dry  pith.     Rub  the  glass  with  silk  and  explain  the 
agitation  of  the  pith  observed. 

6.  Balance  a  meter  stick  on  the  smooth  bottom  of  a  flask  and  see 
if  it  is  affected  by  a  charged  glass  rod.     Are  only  very  light  bodies 
attracted? 


340 


A   HIGH   SCHOOL  COURSE   IN   PHYSICS 


2.   ELECTRIC  FIELDS   AND  ELECTROSTATIC  INDUCTION 

351.  Lines  of  Force.  — We  have  already  observed  that 
an  electrified  body  has  an  effect  upon  another  body  even 
when  placed  at  a  distance  of  several  inches.     Light  objects 
will  rise  from  the  table  toward  a  charged  glass  rod,  and 
the  leaves  of  an  electroscope  will  often  show  a  divergence 

at  a  distance  of  four  or  five 
feet  from  the  charge.  The 
space  in  which  an  electric 
charge  affects  surrounding 
objects  is  called  the  electric 
field  due  to  that  charge. 
If  lines  are  drawn  in  an 
electric  field  to  represent 
at  every  point  the  direction 
in  which  a  charge,  as  A, 

P.O.  276. -Lines  of  Force  Emanating      Fig'     276'    tends    tO .  mOV6 

from  a  Charge.  a    very    small,    positively 

charged  body  B,  they  are  called  lines  of  force.     Thus  every 
electric  field  is  assumed  to  be  filled  with  lines  of  force. 

Lines  of  force  extend  from  positive  charges  to  nega- 
tive charges,  as  shown  in  Fig.  277.  In  other  words,  a 
positive  charge  always  exists  at 
one  end  of  a  line  of  force  and 
a  negative  one  at  the  other.  „ 
Therefore,  since  each  line  of  / 
force  must  have  two  extremi- 
ties, for  every  positive  charge  FIG.  277.  —  The  Electric  Field  be- 
,,  ,  ,  ,  tween  Two  Unlike  Charges. 

there  must  be  somewhere  a  cor- 
responding negative  charge.     This  fact  gives  rise  to  'the 
phenomena  shown  in  the  following  sections. 

352.  Electrification  by  Induction.  —  Let  a  charged  glass  rod 
be  brought  near  a  gold-leaf  electroscope.     The  leaves  begin  to  diverge 


ELECTROSTATICS 


341 


when  the  rod  is  at  a  distance  and  spread  more  and  more  as  the  rod 
approaches.     On  removing  the  rod  the  leaves  fall  together  again. 

The  temporary  effect  produced  in  the  electroscope,  due 
to  the  presence  of  a  neighboring  charge,  is  the  result  of 
electrostatic  induction.  The  leaves  of  the  instrument  are 
obviously  affected  only  while  they  are  in  the  electrical 
field  around  the  charged  rod,  since  they  collapse  as  soon 
as  the  rod  is  removed.  In  this  case  there  is  no  change  in 
the  amount  of  electrification  on  the  glass  rod,  and  the 
effect  can  easily  be  shown  to  take  place  through  an  inter- 
vening plate  of  glass  or  other  insulating  material.  Hence, 
in  the  process  of  electrostatic  induction  a  transfer  of  elec- 
tricity from  one  body  to  the  other  does  not  take  place. 

353.  Electrical  Separation  by  Induction.  —  The  condition 
of  an  electroscope  which  is  placed  near  a  charged  body,  as 
in  the  preceding  section,  is  readily  understood  after  the 
following  experiment  f 

Place  two  metal  vessels  A  and  B,  Fig.  278,  in  contact  on  an  elevated 
block  of  paraffin,  or  some  other  good  insulating  material.  Each  ves- 
sel should  be  provided  with 
a  small  pith  ball  attached 
to  the  top  by  means  of  a 
piece  of  cotton  thread.  Now 
bring  a  positively  charged 
glass  rod  R  near  one  of  the 
vessels,  as  B,  and,  by  means 
of  the  silk  thread  C,  sepa- 
rate the  vessels  before  re- 
moving the  rod.  When  the 
rod  is  taken  away,  the  pith 
balls  will  show  that  each 


FIG.  278.  —  Separation  of  Positive  and 
Negative  Charges. 


vessel  has  acquired  a  charge,  but  the  charge  on  the  rod  R  has  not  been 
diminished.  With  the  help  of  the  proof  plane  and  electroscope  test 
each  charge  ;  that  of  B  will  be  negative,  that  of  A  positive. 

From  this  experiment  it  is  clear  that  whenever  an  electri- 
fied body  is  brought  near  an  unelectrified  insulated  conductor, 


342          A  HIGH   SCHOOL   COURSE   IN    PHYSICS 

the  opposite  kind  of  electrification  is  induced  on  the  nearer 
side  and  the  same  kind  on  the  remote  side. 

354.  Charging  by  Induction.  —  An   insulated  metallic 
body  may  be  charged  by  making  use  of  the  separation  of 
positive  and  negative  electricity,  as  shown  in  the  preceding 
section. 

Hold  the  positively  charged  glass  rod  about  6  in.  from  the  plate  of 

the  gold-leaf  electroscope.  The  leaves 
diverge  because  the  positive  charge 

--H -4- ,u          "    "%     ^C-      on  the  r°d  induces  a  negative  charge 

in  the  plate  and  a  positive  charge  in 
the  leaves,  as  shown  in  Fig.  279. 
While  the  rod  is  in  this  position, 
touch  the  electroscope  with  the  finger. 

•f  /  \  +  The  leaves  collapse.     Now  remove  the 

FIG.  279.  —Charged  RodjVcting     finger  and  then  the  rod.     The  leaves 

diverge  again,  thus  showing  that  a 
charge  is  left  on  the  instrument.  If 
this  charge  be  tested  by  bringing  a  negative  charge  near  the  elec- 
troscope, an  increased  divergence  will  show  that  it  is  negative. 

355.  Explanation  of  the  Process. — The  electrical  con- 
ditions that  exist  during  the  process  of  charging  a  conduc- 
tor by  induction  are  represented  in  Fig.  280.     The  presence 


(I)  (2)  (3) 

FIG.  280.-  Illustrating  the  Process  of  Charging  by  Induction. 

of  the  positively  charged  rod  R  produces  a  separation  of 
positive  and  negative  electricity  in  the  conductor  A  (§  353), 
just  as  if  an  uncharged  body  possessed  equal  amounts  of 
these  two  kinds,  as  shown  in  (1).  Lines  of  force  extend 
from  the  positive  on  the  rod  to  the  induced  negative  on 


ELECTROSTATICS 


343 


the  conductor,  while  other  lines  extend  away  from  the 
positive  at  the  remote  end  to  the  walls  of  the  room, 
near-by  objects,  etc.  When  the  conductor  is  touched  with 
the  finger,  as  in  (2),  the  repelled  or  "free"  charge,  which 
in  this  case  is  positive,  is  permitted  to  escape  through  the 
body  to  the  earth.  The  negative  or  "  bound  "  charge  re- 
mains on  the  conductor  on  account  of  the  influence  of  the 
charged  rod  R.  On  removing  first  the  finger  and  then 
the  rod,  the  negative  charge 
distributes  itself  over  the  con- 
ductor, as  shown  in  (3).  If  R 
is  a  negatively  charged  body, 
the  signs  of  all  the  charges  will 
simply  be  reversed. 

356.  Distribution  of  Electric- 
ity on  a  Conductor.  —  Insulate  a 
metal  vessel  A,  Fig.  281,  by  placing 
it  on  a  dry  tumbler  B,  or  a  plate  of 
paraffin  or  beeswax.  Charge  the  ves- 
sel as  highly  as  possible  and  then 
try  to  take  charges  from  its  interior 
surface  to  the  electroscope  by  means 
of  a  proof  plane  c.  It  will  be  found 
that  no  charge  can  be  obtained  in 
this  way.  Now  try  to  take  a  charge 


Fia.  281.  — No  Charge  can  be 
Taken  from  the  Interior  Sur- 
face of  a  Charged  Body. 


from  the  exterior  surface  of  the  vessel.     The  electroscope  will  show 
that  this  attempt  is  successful. 

This  experiment  shows  very  conclusively  that  an  elec- 
trical charge  distributes  itself  over  the  exterior  surface  of 
an  insulated  conductor.  This  is  just  the  result  that  is  to 
be  expected  if  we  consider  that  the  various  parts  of  any 
charge  are  mutually  repellent.  On  this  account  they  will 
separate  to  the  greatest  extent  possible,  which  is  the  case 
when  the  charge  is  distributed  over  the  exterior  surface  of 
the  body  charged. 


344          A  HIGH   SCHOOL   COURSE   IN   PHYSICS 

357.  Distribution  of  a  Charge  not  Uniform.  —  Charge  a  large 
insulated  egg-shaped  conductor,  Fig.  282.     By  means  of  a  proof  plane 

transfer  a  charge  from  the  large  end  of  the  body  to 
the  electroscope  and  observe  the  divergence  of  the 
leaves.  Now  discharge  the  electroscope  and  test  the 
small  end  of  the  conductor  in  a  similar  manner.  A 
greater  divergence  of  the  leaves  will  be  obtained. 

It  thus  appears  that  an  electrical  charge 
"IGtricai~Den-     distributes  itself  over  a  conducting  surface 
sity  is  Great-    in  accordance  with  the  shape  of  the  body. 
Printed  End     The    quantity    per    square    centimeter    is 
of  this  Con-    greatest  at  the  small  end.     In  other  words, 
the  electrical  density  is  greatest  where  the 
conductor  is  the  most  sharply  pointed.     In  fact,  if  a  sharp 
point  be  attached  to  the  charged  conductor,  the  density  at 
the  point  may  be  so  great  that  the  charge  will  be  sponta- 
neously discharged  into  the  surrounding  air. 

358.  Effect  of  Points.  —  By  the  help  of  sealing  wax  or  shellac 
attach  the  center  of  a  sharp  needle  to  a  glass  handle.     Bring  a  charged 
glass  rod  over  the  electroscope,  and,  while  the  rod  remains  in  this 
position,  bring  the  eye  end  of  the  needle  against  the   plate  of  the 
electroscope  and  the  point  toward  the  charged  rod.     Now  remove 
both  needle  and  rod,  and  the  electroscope  will  be  positively  charged. 
Again,  hold  a  charged  glass  rod  near  an  insulated  conductor,  as  a 
metal  vessel,  and  then  place  the  eye  end  of  the  needle  against  the 
opposite  side  of  the  conductor.     Remove  the  needle  and  then  the  rod 
and  test  the  conductor  for  a  charge.     A  negative  charge  will  be  found. 

The  results  of  these  experiments  depend  upon  the  electri- 
cal discharge  from  pointed  conductors.  On  account  of 
the  great  electrical  density  at  a  point,  an  intense  field  of 
force  exists  in  the  immediate  neighborhood.  Air  particles 
are  forcibly  drawn  against  the  point  and  charged  by  con- 
tact with  the  same  kind  of  electricity,  after  which  they  are 
violently  repelled.  Thus  a  so-called  electrical  wind  is  set 
up  which  conveys  away  the  charge  at  a  rapid  rate. 


ELECTROSTATICS  3^5 

359.  Lightning  Rods.  — Positive  evidence  regarding  the 
identity  of  lightning  and  electrical  discharges  was  secured 
by  the  classical  experiments  of  Benjamin  Franklin1  in  1752, 
when  he  succeeded  in  drawing  an  electrical  charge  from  a 
thunder  cloud  along  the  string  of  a  kite.  Through  his 
suggestion  lightning  rods  were  first  used  as  a  protection  for 
buildings  against  damage  by  lightning.  The  principle  has 
been  demonstrated  in  the  preceding  section.  A  strongly 
charged  cloud  passes  over  a  building,  and  between  the  cloud 
and  the  building  there  is  set  up  an  intense  field  of  force 
(§  851).  If  this  field  becomes  of  sufficient  intensity,  the 
air,  being  no  longer  able  to  insulate,  breaks  down  as  an 
insulator.  The  sudden  discharge  tears  the  roof  of  the 
building,  and  the  intense  heat  often  produces  a  conflagra- 
tion. The  presence  of  a  pointed  conductor,  however,  lead- 
ing from  above  the  roof  to  the  earth,  permits  a  gentle  and 
harmless  discharge  of  jelectricity  to  take  place  between  the 
cloud  and  the  earth.  The  effectiveness  of  lightning  rods 
is  often  impaired  by  the  use  of  dull  points  and  poor  ground 
connections. 

The  thunder  that  accompanies  a  lightning  flash  is  caused 
by  the  impact  of  the  air  as  it  is  forced  in  to  fill  the  partial 
vacuum  which  is  developed  along  the  line  of  electrical  dis- 
charge. At  a  distance  from  the  discharge,  the  direct 
report  is  followed  by  echoes  from  the  clouds  and  woods, 
causing  the  rumblings  so  common  in  most  localities. 

EXERCISES 

1.  In  testing  a  charge,  why  is  it  necessary  to  work  with  a  charged 
electroscope  ? 

2.  AVhen  a  gold-leaf  electroscope  is  charged  with  negative  elec- 
tricity, for  example,  why  will  the  approach  of  a  positively  charged 
body  produce  first  a  decrease  and  then  an  increase  of  the  divergence 
of  the  leaves? 

1  See  portrait  facing  page  346. 


346 


A   HIGH   SCHOOL  COURSE   IN   PHYSICS 


3.  If  several  insulated  metal  vessels  are  placed  in  a  row,  but  not  in 
contact  with  each  other,  what  will  be  the  electrical  condition  of  each 
when  a  positively  charged  rod  is  held  near  one  end  of  the  row?    Illus- 
trate by  means  of  a  diagram  showing  four  such  vessels. 

4.  If  the  vessel  at  the  end  of  the  row  near  the  charged  rod  is 
touched  with  the  finger  and  then  both  finger  and  rod  are  removed, 
what  will  be  the  electrical  condition  of  each  vessel  ?    Illustrate  this 
case  by  a  diagram. 

5.  Represent  by  diagrams  the  fields  of  force  existing  under  the 
conditions  described  in  Exer.  3. 

6.  Represent  the  fields  of  force  as  they  exist  under  the  conditions 
given  in  Exer.  4  and  compare  each  field  with  the  corresponding  one 
of  the  preceding  exercise. 

3.    POTENTIAL   DIFFERENCE    AND    CAPACITY 

360.  Electrical  Flow.  —  It  has  already  been  observed  in 
§  349  that  a  charge  of  electricity  can  be  transferred  by  a 
conductor  from  an  electrified  body  to  the  electroscope. 
The  effects  that  accompany  such  a  transmission  of  elec- 
tricity, as  will  be  seen  later,  enable  it  to  be  employed  as 
one  of  the  most  important  agents  under  the  control  of  man. 
The  following  experiment  will  bring  out  more  clearly  the 

conditions  under  which  a 
transfer  of  electricity  takes 
place. 

Provide  two  similar  metal  ves- 
sels A  and  B,  Fig.  283,  with  pith 
balls  and  insulate  them  on  blocks 
of  paraffin.  Let  each  vessel  be 


positively  charged,  but  A  more 
highly  than  B,  as  measured  by  the 
repulsion  of  the  pith  balls.  Now 
connect  the  vessels  by  means  of  a  wire  attached  to  a  sealing  wax 
handle  C.  The  pith  ball  on  A  will  fall  slightly  while  that  on  B  will 
rise.  Thus  some  of  the  charge  on  A  moves  along  the  wire  to  B. 


FIG.  283.  —  Conductor  A  has  a  Higher 
Potential  than  B. 


Although  both  insulated  vessels  were  originally  charged 
with  the   same   kind  of   electricity,  in  the   language    of 


BENJAMIN    FRANKLIN    (1706-1790) 

The  achievements  of  Franklin  in  the  field  of  electricity  are  no 
less  brilliant  than  his  successes  as  a  statesman  and  diplomat.  His 
attention  was  first  turned  to  the  study  of  electrical  phenomena  by 
witnessing  some  experiments  which  at  that  time  (1746)  were  re- 
garded as  no  less  than  marvelous.  His  experiment  to  prove  the 
electrical  nature  of  lightning  has  become  classic.  In  order  to  ascer- 
tain whether  electricity  could  be  obtained  from  clouds  during  a  storm, 
Franklin  constructed  a  kite  which  was  provided  with  a  pointed  metal 
rod  for  "  drawing  off "  the  electrical  charge,  if  there  should  prove 
to  be  any.  At  the  approach  of  a  storm,  Franklin  and  his  son  raised 
the  kite,  which  was  held  by  a  hempen  cord.  The  lower  end  of  the 
cord  was  tied  to  a  metal  key  through  which  was  passed  a  ribbon 
of  silk  to  protect  the  body  from  severe  and  dangerous  shocks.  As 
soon  as  the  cord  became  moistened  by  the  rain,  electric  sparks  were 
readily  drawn  from  the  key.  In  regard  to  this  experiment,  Franklin 
writes:  "At  the  key  the  Leyden  jar  may  be  charged;  and,  from  the 
electric  fire  thus  obtained,  spirits  may  be  kindled  and  all  other 
electrical  experiments  performed  which  are  usually  done  by  the 
help  of  a  rubbed  glass  globe  or  tube,  and  thereby  the  sameness  of 
the  electrical  matter  with  that  of  lightning  completely  demonstrated." 

"  Antiquity  would  have  erected  altars  to  this  great  and  powerful 
genius  who,  to  promote  the  welfare  of  mankind,  comprehending  both 
the  heavens  and  the  earth  in  the  range  of  his  thought,  could  at  once 
snatch  the  bolt  from  the  cloud  and  the  scepter  from  tyrants." — 
MIRABEAU 


ELECTROSTATICS 


347 


Physics,  A  was  charged  to  a  higher  potential  than  B.  Thus 
a  charge  moves  from  a  point  of  higher  to  one  of  lower  poten- 
tial, just  as  heat  flows  along  a  heat  conductor  from  a  point 
of  higher  to  one  of  lower  temperature. 

361.  Potential  Difference.  —  It  has  just  been  observed 
that  an  electrical  charge  will  flow  along  a  conductor  as 
long  as  there  exists  a  potential  difference.  It  is  precisely 
this  difference  of  poten- 
tial that  determines 
whether  electricity  will 
flow  and  the  direction 
it  will  take.  Figure 
284  will  show  to  some 
extent  the  differences 
that  may  exist  between 
electrical  charges.  AB 
represents  the  levei 
ground,  and  a,  b,  <?,  and 
d  four  tanks  containing  water.  Tank  a  is  analogous  to 
an  insulated  body  charged  positively  to  a  high  potential, 
as  vessel  a',  while  tank  b  represents  one  of  a  lower  poten- 
tial than  af,  as  b' .  It  is  assumed  that  all  positive  charges 
are  of  a  higher  potential  than  the  earth  and  will  discharge 
to  the  earth  unless  insulated  from  it.  On  the  other  hand, 
negative  charges,  as  cf  and  df,  are  assumed  to  have  lower 
potentials  than  the  earth.  The  potential  of  the  earth  is 
thus  on  the  dividing  line  between  positive  and  negative 
charges;  hence,  the  potential  of  the  earth  is  regarded  as 
zero. 

Connecting  any  two  of  the  tanks  shown  in  the  figure 
would  evidently  result  in  a  transfer  of  water  from  left  to 
right ;  thus  joining  any  two  of  the  charged  vessels  by 
means  of  a  conductor  would  bring  about  a  transfer  of 
electricity  in  the  same  direction. 


FIG.    284. —  Analogy   between    Electrical 
Potential  and  Water  Level. 


318          A   HIGH   SCHOOL   COURSE   IN    PHYSICS 

362.  Mixing  Positive  and  Negative  Charges.  —  Let  the  two 

insulated  metal  vessels  used  in  §  360  be  charged  with  equal  amounts 
of  positive  and  negative  electricity  as  shown  by  the  divergence  of  the 
pith  balls.  Now  connect  them  by  means  of  a  wire  provided  with  an 
insulating  handle  as  before.  Both  balls  fall  against  the  sides  of  the 
vessels,  showing  that  the  two  charges  have  neutralized  each  other. 
Repeat  the  operations  just  described  after  charging  A  positively  until 
its  pith  ball  diverges  considerably  more  than  that  of  B,  which  is 
charged  negatively.  After  the  connection  has  been  made  between 
the  vessels,  a  positive  charge  will  be  found  on  each  vessel. 

Not  only  does  electricity  tend  to  flow  from  a  positively 
to  a  negatively  charged  body,  but  a  mixture  of  unlike 
charges  tends  to  produce  a  cancellation  of  both.  If,  how- 
ever, one  of  the  charges  exceeds  the  other  in  amount,  the 
excess  remains  distributed  over  both  surfaces.  In  the 
final  condition  both  bodies  are  of  the  same  potential. 

363.  The  Electrostatic  Unit  of  Quantity.  —  Unit  charges 
of  electricity  are  such  equal  quantities  as  exert  upon  each 
other  a  force   of  one  dyne  (§  35)  when  separated  by  one 
centimeter  of  air.     The  force,  as  we  have  seen,  is  repellent 
when  the  charges  are  of  the  same  kind  and  attractive 
when  they  are  different.     If  two  equal  quantities  of  posi- 
tive and  negative  electricity  are  mixed,  they  cancel  each 
other;  but  if  both  of  the  two  mixed  charges  are  of  the 
same  kind,  the  resulting  charge  is  their  sum. 

EXAMPLE.  —  If  10  units  of  positive  electricity  are  mixed  with  12 
units  of  negative,  what  will  be  the  result? 

SOLUTION.  —  The  10  positive  units  will  cancel  10  negative  units, 
leaving  2  units  of  negative  electricity,  which  will  be  distributed 
over  the  surface  of  the  conductors. 

364.  Electrostatic    Capacity.  —  By   means   of  a  proof   plane 
transfer  a  charge  from  an  electrified  glass  rod  to  the  electroscope  and 
note  the  approximate  divergence  of  the  leaves.     Now  place  one  of  the 
metal  vessels  used  in  §  353  upon  the  plate  of  the  electroscope  and  again 
transfer  a  charge  with  the  proof  plane  as  before.     In  this  case  the 
divergence  will  be  found  to  be  much  smaller  than  before.     Continue 
to  transfer  charges  until  the  divergence  is  the  same  as  at  first. 


ELECTROSTATICS 


349 


By  placing  the  metal  vessel  upon  the  electroscope,  the 
surface  over  which  a  charge  will  distribute  itself  is 
materially  increased.  Hence  a  given  quantity  of  elec- 
tricity will  produce  a  smaller  electric  density,  and  the 
mutual  repulsion  of  the  gold  leaves  will  be  lessened.  On 
this  account,  a  larger  quantity  will  be  required  to  produce 
the  same  divergence  of  leaves  ;  or,  in  other  words,  to  produce 
the  same  potential  as  at  first.  The  change  produced  in  the 
condition  of  the  conductor  is  expressed  by  saying  that  the 
electrostatic  capacity  is  increased  by  the  increased  area 
presented  by  the  metal  vessel. 

365.  Condensers. — In  many  of  the  practical  applica- 
tions of  electricity,  it  is  necessary  to  make  use  of  some 
device  that  has  many  times  the  capacity  of  anything  we 
have  used  in  the  preceding  experiments.  The  manner  in 
which  the  desired  result  is  accomplished  is  made  clear  by 
the  following  experiments : 

1.  Place  a  flat  metal  plate  (7,  Fig.  285,  having  well-rounded  cor- 
ners, upon  the  plate  of  the  gold-leaf  electroscope  and  charge  the  in- 
strument with  the  glass  rod  and  proof  plane.     Now  bring  a  similar 
metal  plate  B,  which  is  provided  with  a  handle 

of  sealing  wax  A,  near  C,  but  not  touching 
it.  The  divergence  of  the  gold  leaves  will 
decrease,  showing  that  the  potential  of  the 
leaves  has  been  lowered.  Withdraw  B  and 
the  potential  of  the  leaves  will  rise  to  its 
original  value.  Repeat  these  operations  while 
the  fingers  are  allowed  to  remain  in  contact 
with  B,  thus  connecting  it  with  the  earth. 
The  divergence  of  the  leaves  becomes  very 
small  when  B  and  C  are  near  together,  but 
returns  to  its  former  value  when  the  plates 
are  again  separated. 

2.  Mount  a  plate  B  in  a  clamp  so  that  it 
is  about  |  cm.  from  C  and  insulated.     Count 
the  number  of  charges  that  must  be  carried 


FIG.  285.  — Showing  the 
Effect  of  a  Neigh- 
boring Conductor  on 
Capacity. 


to  C  by  a  proof  plane  to  produce  a  given  divergence,  i.e.  to  produce  a 


350          A  HIGH   SCHOOL   COURSE   IN   PHYSICS 

given  potential.  Now  discharge  C  and  connect  B  with  the  earth  by 
means  of  a  wire,  and  see  how  many  charges  must  be  transferred 
to  C  to  produce  approximately  the  same  divergence  as  before.  A 
large  number  will  be  required. 

These  experiments  show  (1)  that  the  potential  that  a 
given  charge  produces  when  placed  on  an  insulated  con- 
ductor depends  upon  the  proximity  of  other  conductors 
and  whether  they  are  "  earthed "  or  not,  and  (2)  that  the 
capacity  of  a  conductor  is  enormously  increased  by  the 
presence  of  an  earthed  conductor  placed  very  near,  but 
insulated  from  it.  A  combination  of  plates  separated  by  an 
insulator  constitutes  an  electrostatic  condenser.  The  capac- 
ity of  a  condenser  is  proportional  to  the  size  of  the  plates 
and  becomes  greater  as  the  distance  between  them  is 
reduced. 

366.  Influence  of  the  Insulating  Material Arrange  the 

electroscope  and  plates  as  in  Experiment  2  of  the  preceding  section. 
Connect  B  with  the  earth  and  charge  C  until  a  moderate  divergence 
of  the  leaves  is  produced.     Now  introduce  between  B  and  C  a  pane  of 
glass  and  note  the  effect  upon  the  divergence.     The  potential  of  C 
will  fall  when  the  glass  is  introduced,  and  will  rise  again  when  it  is 
removed.     Let  the  experiment  be  made  by  using  beeswax  or  paraffin 
instead  of  glass.     The  effect  is  more  marked  than  before. 

Since  the  introduction  of  another  insulator,  or  dielectric, 
to  replace  the  air  between  the  plates  of  a  condenser,  re- 
duces the  potential,  it  is  obvious  that  it  will  require  a 
greater  charge  to  bring  the  potential  back  to  its  original 
value.  Hence  the  capacity  of  the  condenser  is  increased. 
One  of  the  best  insulators  used  in  condensers  is  mica,  not 
only  because  it  can  easily  be  obtained  in  thin  sheets,  but 
because  of  its  advantageous  influence  as  a  dielectric  upon 
the  electrical  capacity  of  the  condenser. 

367.  Forms  of  Condensers.  —  One  of  the  earliest  forms 
of  condensers  is  known  as  the  Ley  den  jar,  from  Ley  den 


ELECTROSTATICS 


351 


FIG.  286.  —The  Ley- 
den  Jar. 


in  Holland,  the  place  of  its  origin.  It  was  first  used  in 
1745.  This  condenser  consists  of  a  glass  jar,  Fig.  286, 
which  is  coated  with  tin  foil  to  about 
two  thirds  of  its  height  on  its  interior 
and  exterior  surfaces.  Through  a  cover 
of  insulating  material  passes  a  metal  rod 
terminating  at  the  top  in  a  ball  and  at 
the  lower  end  in  a  chain  which  makes 
contact  with  the  inner  coating  of  the  jar. 
Another  form  of  condenser  that  is 
widely  used  is  represented  diagrammat- 
ically  in  Fig.  287.  This  condenser  con- 
sists of  a  large  number  of  sheets  of  tin 
foil  of  which  alternate  sheets  are  con- 
nected at  A  and  the  intervening  sheets  at  B.  In  the 
best  condensers  of  this  type  the  insulating  material  sepa- 
rating the  sheets  of  foil  is  mica;  but  in  cheaper  forms, 
paraffined  paper  is  used.  The  capacity 
of  such  condensers  is  large  on  account 
of  the  large  area  of  tin  foil  and  the 
extremely  small  distance  between  the 
conducting  surfaces  of  the  foil  (§  365). 
368.  Charging  and  Discharging  a  Ley- 
den  Jar.  —  In  order  to  charge  a  Ley- 
den  jar,  the  outer  coating  is  connected  with  the  earth  by 
a  metallic  conductor,  or  the  jar  is  simply  held  in  the  hand. 
A  very  imperfect  earth  connection  is  produced  by  setting 
the  jar  upon  a  table  oi/dry  wood.  If  now  the  knob  of  the 
jar  be  connected  with  some  source  of  electricity,  a  charge 
is  communicated  to  the  inner  surface.  If  this  is  positive, 
an  equal  amount  of  negative  will  be  induced  on  the  inner 
surface  of  the  outer  coating,  and  a  similar  quantity  of 
positive  repelled  to  the  earth.  In  this  manner  a  strain 
is  set  up  in  the  glass  which  in  some  instances  is  suffi- 
cient to  break  the  jar. 


FIG.  287.  — Diagram  of 
a  Mica,  or  Paper, 
Condenser. 


352          A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

To  discharge  a  Leyden  jar,  it  is  necessary  to  make  an 
electrical  connection  between  the  two  tin-foil  coatings. 
If  the  charge  is  not  large,  this  can  be  done  safely  by 
simultaneously  touching  the  knob  and  the  outer  coating 
with  the  hands.  The  best  method,  however,  is  to  bend  a 
metal  conductor  in  such  a  form  that  one  end  can  be  kept 
in  contact  with  the  outer  coating  while  the  other  is  brought 
near  the  knob.  At  the  instant  of  discharge  a  bright  spark 
will  be  seen  to  jump  between  the  knob  and  the  end  of  the 
conductor. 

EXERCISES 

1.  Three  Leyden  jars  are  charged  with  —  4,  —  7,  and  +  10  units 
respectively.     How  many  units  will  remain  in  the  jars  after  the  knobs 
have  been  connected? 

2.  A  Leyden  jar  that  is  placed  on  a  plate  of  glass  has  a  small 
capacity.     Why  ? 

3.  If  a  Leyden  jar  be  highly  charged  and  then  placed  on  a  plate 
of  glass  or  paraffin,  the  knob  can  be  safely  touched  with  the  hand. 
In  this  condition  only  a  small  portion  of  the  entire  charge  will  be 
taken  from  the  jar.     Explain. 

4.  Having   a   metal  globe   positively   electrified,  how  could  you 
electrify  any  number  of  other  globes  with  negative  electricity  ? 

5.  With   a   positively  charged   globe,  how  could  you  positively 
charge  one  of  the  other  globes  without  reducing  the  charge  on  the 
first? 

4.    ELECTRICAL   GENERATORS 

369.    The  Electrophorus.  —  In  order  to  produce  larger 
electrical  charges  than  those  used  in  any  of  the  preceding 
experiments,    the  principle   of   induction 
(§    352)    is    advantageously    employed. 
The  simplest  form  of  generator  for  this 
purpose  is  the  electrophorus  (pronounced 
e  lek  trof'o  rus),   an   apparatus   invented 
FIG.    288.  — The     by  Volta,1  an  Italian  physicist,  in  1777. 
Electrophorus.        XMS   instrument   consists   of   a  plate  of 

1  See  portrait  facing  page  352. 


COUNT    ALESSANDRO    VOLTA    (1745-1827) 

The  age  of  practical  electricity  began  with  the  invention  of  the 
voltaic  cell  by  Volta,  an  Italian,  professor  of  physics  at  the  Uni- 
versity of  Pavia.  Before  his  invention  it  was  not  known  that  a 
continuous  current  of  electricity  could  be  produced.  In  1793,  Volta 
announced  to  the  Royal  Society  of  London  a  discovery  made  by 
Galvani  (1737-1798),  professor  of  anatomy  at  Bologna.  Galvani 
had  observed  that  by  joining  together  two  different  metals  and 
touching  one  to  the  muscle  of  a  frog's  leg  and  the  other  to  the 
nerve,  violent  contractions  were  produced  even  after  the  animal's 
death.  Most  extravagant  hopes  were  founded  on  this  discovery,  and 
many  believed  that  a  cure  had  been  found  for  all  diseases.  In 
reality,  the  discovery  paved  the  way  to  other  and  greater  ones  which 
have  become  a  vital  part  of  the  history  of  the  nineteenth  century. 

The  use  of  the  two  metals  in  Galvani's  experiment  suggested  to 
Volta  the  basic  principle  of  the  modern  cell  consisting  of  two  plates 
of  different  metals  immersed  in  a  liquid.  This,  as  we  know,  becomes 
a  continuous  source  of  electricity.  The  invention  was  received  with 
great  enthusiasm.  In  a  short  time  batteries  made  by  joining  several 
of  the  so-called  "  voltaic  piles  "  were  used  in  experimental  work  in 
many  of  the  laboratories  of  Europe.  With  the  advent  of  this  means 
of  generating  electric  currents  began  the  series  of  discoveries  that 
have  led  up  to  the  phenomenal  use  of  electricity  at  the  present  day. 

To  Volta  is  ascribed  the  invention  of  the  electrophorus,  electro- 
scope, and  the  condenser.  The  practical  unit  of  potential  difference 
is  called  the  volt  in  his  honor. 


ELECTROSTATICS  353 

ebonite  or  a  shallow  metal  dish  B,  Fig.  288,  about  25  cm.  in 
diameter,  into  which  has  been  poured  melted  resin  or 
shellac,  and  a  flat  circular  metal  disk  J.,  somewhat  smaller 
than  the  plate  and  having  well-rounded  edges.  The  metal 
disk  is  provided  with  an  insulating  handle  of  ebonite. 

Let  the  plate  of  an  electrophorus  be  stroked  with  cat's  fur  and  its 
electrification  tested.  Place  the  metal  disk  upon  the  plate  and  test 
the  kind  of  electricity  that  can  be  taken  from  its  upper  surface. 
Touch  the  disk  with  the  finger  in  order  to  "ground"  or  "earth"  it, 
lift  it  by  the  insulating  handle,  and  test  its  charge.  Repeat  the  ex- 
periment, without  "  grounding  "  the  disk.  Explain  the  result. 

The  action  of  the  electrophorus  is  as  follows  :  When 
the  non-conducting  plate  B  is  stroked  with  fur,  it  is  given 
a  negative  charge.  If,  now,  the  metal  n 

disk  A  be  placed  upon  the  plate,  the 
negative  on  the  plate  induces  a  posi- 
tive charge  on  the  lower  surface  of  the       \^  ±  ±  ±~^| 
disk  and  repels  a  negative  charge  to  n 

the  upper  side,  as  shown  in  Fig.  289. 
When   the   disk   is  touched   with  the         (+  +  +  +  +^A 
finger,    the   negative    charge    escapes, 
leaving  the  positive  charge,   which  is 


-,.,.,  ,  ^         v   i          i  -,.    •        FlG-   289.  —  Represent- 

aistributecl    over  the  disk  when  it  is         ing  the  Theory  of 
lifted.     Compare  §  355.     Any  number         the  Electrophorus. 
of  charges  may  be  obtained  from  the  electrophorus  with- 
out producing  any  appreciable  change  in  the  charge  on 
the  plate. 

370.  Source  of  the  Energy  Derived  from  the  Electrophorus. 
—  Although  any  number  of  charges  can  be  produced  by 
the  electrophorus,  we  know  that  the  energy  obtained  in 
this  manner  cannot  be  brought  into  existence  without  the 
expenditure  of  a  like  quantity  on  the  part  of  some  work- 
ing agent  (§  64).  The  source  of  the  energy  is  obvious 

from  the  following  : 
24 


354 


A  HIGH   SCHOOL  COURSE  IN   PHYSICS 


After  the  metal  disk  of  an  electrophorus  has  been  placed  on  the 
electrified  plate  and  touched  on  its  upper  surface  to  draw  away 
the  negative  charge  (§  369),  connect  a  gold-leaf  electroscope  with  the 
disk  by  means  of  a  slender  wire.  No  divergence  in  the  leaves  is  pro- 
duced, although  we  know  that  the  disk  has  a  positive  charge  induced 
by  the  negative  charge  on  the  plate.  Slowly  lift  the  disk  by  the 
insulating  handle  and  the  leaves  will  diverge  widely. 

When  the  disk  with  its  induced  positive  charge  rests 
upon  the  charged  plate,  it  is  at  zero  potential  because  it 
has  been  in  connection  with  the  earth.  The  disk,  like  a 
weight  lying  on  the  ground,  manifests  no  energy  until  its 
positive  charge  is  separated  from  the  negative  charge  on 
the  plate  in  opposition  to  their  attractive  force.  The 
agent,  therefore,  that  lifts  the  disk  from  the  plate  furnishes 
the  energy  which  appears  in  the  disk  as  electrical  energy. 
When  the  disk  is  discharged,  the  electrical  energy  is 
transformed  into  heat  which  appears  in  the  spark  that  is 
produced.  The  heat  of  the  spark  can  be  utilized  to  light 
illuminating  gas  or  explode  powder. 

371.  The  Toeppler-Holtz  Influence  Machine. — The  dis- 
covery of  X-rays  and  their  necessity  for  generators  capa- 


FIG.  290.  —  The  Toeppler-Holtz  Electrical  Machine. 


ELECTROSTATICS  355 

ble  of  producing  a  continuous  supply  of  electricity  has 
brought  such  generators  into  extensive  use.  The  simplest 
form  of  induction,  or  influence,  machines  is  the  Toeppler- 
Holtz,  shown  in  Fig.  290.  This  machine  makes  use  of  a 
glass  plate  about  50  cm.  in  diameter,  which  is  provided 
with  six  or  eight  metallic  disks,  as  shown.  This  plate 
revolves  in  front  of  a  second  stationary  plate  of  glass  P, 
which  is  furnished  with  two  strips  of  tin  foil  6,  6',  covered 
with  paper  sectors  called  arma- 
tures, or  inductors.  In  front 
of  the  revolving  plate  is  the 
stationary  neutralizing  bar  J5, 
provided  with  points  and  tinsel 
brushes  at  the  ends.  (7,  C'  are 
two  conductors  having  at  their 
other  extremes  points  that  come 
close  to  the  revolving  plate.  _ 

The  action   of  the  machine  is    FlG<  391. -Diagram  Showing 

best  Understood  from  a  Study  of  Action  of  the  Toeppler-Holtz 

the  diagram  shown  in  Fig.  291. 

Imagine  a  small  positive  charge  placed  on  the  sector  b. 
This  charge  acts  inductively  through  the  glass  on  the  rod 
BB',  attracting  a  negative  charge  to  B  and  repelling  a 
positive  one  to  B1 '.  These  induced  charges  rapidly  escape 
from  the  points  (§  358)  and  electrify  the  glass  plate  as  well 
as  each  metallic  disk  as  it  passes.  As  the  plate  rotates  in 
the  direction  shown  by  the  arrow,  the  disks  at  the  top 
carry  negative  charges  to  the  right,  while  those  at  the  bot- 
tom carry  positive  charges  to  the  left.  These  disks  touch 
the  small  brushes  c  and  c',  through  which  negative  charges 
are  given  to  sector  bf  and  positive  to  b.  The  charges  on  the 
sectors  are  thus  continually  increased,  an  effect  which  con- 
tinually increases  the  first  inductive  action  through  the 
glass  on  BB' .  Again,  when  the  negatively  charged  glass 


356          A   HIGH   SCHOOL   COURSE   IN   PHYSICS 

plate  reaches  Q\  positive  electricity  is  drawn  from  the 
points  and  negative  left  on  the  ball  0'.  Similarly,  on 
the  opposite  side  of  the  machine,  the  positively  electrified 
plate  neutralizes  the  negative  charge  that  it  induces  at 
the  points  (7,  and  thus  leaves  a  positive  charge  on  ball  0. 
The  unlike  charges  on  0  and  0'  continue  to  increase  until 
the  difference  of  potential  is  sufficient  to  force  a  dis- 
charge to  take  place  through  the  air  between  them.  If 
the  distance  is  not  too  great,  a  stream  of  sparks  will  appear 
to  pass  without  interruption.  The  energy  transformed  in 
each  spark  is  greatly  increased  by  the  presence  of  the  two 
Leyden  jars  shown  in  Fig.  290. 

EXERCISES 

1.  Why  is  the  metal  disk  of  the  electrophorus  not  charged  nega- 
tively by  contact  with  the  negatively  charged  plate? 

SUGGESTION.  —  Consider  the  nature  of  the  plate  and  the  fact  that  it 
touches  the  disk  in  only  a  very  few  places. 

2.  Explain  why  an  influence  machine  turns  with  greater  difficulty 
when  it  is  developing  a  high  potential  between  the  balls  0  and  O'. 

SUGGESTION.  —  Consider  the  nature  of  the  electrical  forces  existing 
between  the  stationary  and  moving  charges. 

3.  Small  Leyden  jars  (see  Fig.  290)  used  in  connection  with  the 
balls  of  the  influence  machine  cause  the  production  of  much  brighter 
sparks  than  would  be  produced  without  them.     Explain. 

4.  With  the  Leyden  jars  removed,  would  the  frequency  with  which 
sparks  pass  between  0  and  0'  be  increased  or  decreased  ? 

SUMMARY 

1.  Many  substances,  as  glass,  ebonite,  sealing  wax,  etc., 
when  rubbed  with  silk,  flannel,  fur,  etc.,  possess  the  prop- 
erty of  attracting  light  objects,  and  are  therefore  said  to 
be  electrified  (§  343). 

2.  Electrical  charges  are  of  two  kinds,  called  positive 
and  negative  charges  (§  344). 


ELECTROSTATICS  35? 

3.  Similar   charges    repel   each   other   and   dissimilar 
charges  attract  (§  345). 

4.  The  electroscope  is  used  to  determine  the  presence 
of  a  charge  and  its  kind  (§  346). 

5.  Equal  amounts  of  positive  and  negative  electricity 
are  always  developed  simultaneously  (§  848). 

6.  Charges  of  electricity  are  transferred  by  conductors, 
as  metals,  carbon,  etc.     A  substance  that  will  not  act  as 
a  conductor  is  called  an  insulator;  such  are  dry  air,  glass, 
mica,  rubber,  shellac,  etc.  (§§  349  and  350). 

7.  Au  electric  field  is  the  space  around  a  charged  body 
within    which    neighboring   bodies   are   affected   by   the 
charge  (§  351). 

8.  The  electrical  effect  of  an  electrified  body  upon  a 
neighboring  conductor  insulated  from  it  is  the  result  of 
electrostatic  induction  (§  352). 

9.  The  effect  of  induction  by  a  given  charge  is  to  cause 
a  dissimilar  kind  of  electricity  to  appear  on  the  nearer 
side  of  an  insulated  conductor  and  the  similar  kind  on  the 
remote  side  (§§  353  to  3.55). 

10.  Charges  of  electricity  distribute  themselves  over 
the  exterior  surfaces  of  conductors.     The  relative  quan- 
tities per  square  centimeter  depend  on  the  curvature  of 
the  surface,  being  greatest  at  places  of  the  greatest  curva- 
ture, as  at  points  (§  356  and  357). 

11.  Charges    are   rapidly   conveyed  away   from  sharp 
points  by  air  particles,  which  become  charged  by  contact 
and  are  then  repelled  (§  358). 

12.  Charges  flow  along  conductors  from  places  of  higher 
to  places  of  lower  potential.     The  potential  of  the  earth  is 
regarded  as  zero  (§  360  and  361). 

13.  The  union  of  positive  and  negative  charges  of  the 
same  size  produces  a  cancellation  of  both  (§  362). 


358          A  HIGH   SCHOOL   COURSE   IN  PHYSICS 

14.  Unit  charges  are  such  equal  quantities  of  electricity 
as  exert  upon  each  other  a  force  of  one  dyne  when  the 
distance  between  them  in  air  is  one  centimeter  (§  363). 

15.  Electrostatic  capacity  is  measured  by  the  number  of 
units  of  electricity  required  to  produce  a  given  potential. 
The  capacities  of  two  insulated  conductors  are  proportional 
to  the  number  of  units  required  to  bring  them  to  the  same 
potential  (§  364). 

16.  The  capacity  of  a  condenser  is  made  very  large  by 
bringing  the  conductor  to  be  charged  near  another  which 
is  connected  with  the  earth,  but  thoroughly  insulated  from 
the  former.     The  most  common  form  of  condenser  is  the 
Leyden  jar  (§§  365  to  368). 

17.  The  process  of  induction  is  employed  in  the  genera- 
tion of  large  electrostatic  charges.     See  the  electrophorus 
(§  369)  and  the  Toeppler-Holtz  machine  (§  371). 


CHAPTER   XVII 
MAGNETISM 

1.  MAGNETS   AND   THEIR   MUTUAL  ACTION 

372.  Production  of  a  Magnet.  —  Magnets  present  at  least 
two  properties  that  are  familiar  to  every  one:  (1)  the  ends 
of  a  magnet  will  pick  up  small  pieces  of  iron,  as  iron  filings, 
etc.,  and  (2)  a  magnet  will  take  a  north-and-south  posi- 
tion when  properly  suspended.     The  following  experiment 
will  serve  to  show  that  an  intimate  relation  exists  between 
electricity  and  magnetism  : 

Break  off  several  pieces  of  watch  spring,  and  balance  each  of  them 
on  the  head  of  a  pin,  as  shewn  in  Fig.  292.  Observe  that  they  will 

remain  indefinitely  in  any  position  and  will  not          ^f^""^ ==» 

pick  up  iron  filings.     Now  wrap  the  pieces  of     *r^ 
spring  in  a  small  sheet  of  paper  and  place  them  f^rfe^ 

within  a  helix,  or  spiral,  made  by  winding  ten 

or  twelve  turns  of  insulated  copper  wire  upon  FIG.  292.  —  Piece  of 
a  lead  pencil.  Discharge  a  Leyden  jar  through  Watch  Spring  Bal- 
the  helix  of  wire  and  then  remove  the  pieces  of 

spring.  They  will  be  found  able  to  pick  up  iron  filings  and  small  sew- 
ing needles ;  and,  when  balanced  on  the  head  of  a  pin,  each  piece  will 
become  stationary  only  when  in  a  north-and-south  position. 

The  experiment  shows  (1)  that  a  moving  charge  of 
electricity  which  flows  in  a  helix  around  a  piece  of  steel 
magnetizes  it,  and  (2)  that  a  magnetized  bar  of  steel  re- 
tains at  least  a  portion  of  the  magnetism  produced  by  the 
electric  flow. 

373.  Natural  Magnets  or  Lodestones.  — In  ancient  times 
iron  was  mined  on  some  of  the  islands  of  the  Mediterranean 
and  along  the  coasts   of  the   ^Egean  sea.     It  was  early 

359 


360         A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

observed   that  an  occasional  piece  of  the  ore,  magnetite 
(chemical  symbol,  Fe3O4),  possessed  the  power  of  attracting 

small  pieces  of  iron  and  also 
imparted  this  property  by 
contact  to  pieces  of  iron  and 
steel.  According  to  some 
writers,  the  word  "  magnet  " 
is  derived  from  Magnesia  in 
Asia  Minor,  a  province  in 
which  magnetic  iron  ore  is  es- 
pecially abundant.  Speci- 
mens of  ore  that  possess  the 
properties  of  magnets  are 
called  lodestones,  or  natural 
magnets.  See  Fig.  293. 

374.    Poles  of  a  Magnet.  — 
If  a  magnet  be  placed  on  a 

FIG.  293.  —  A  Natural  Magnet. 

sheet  of  paper   and   covered 

with  iron  filings,  it  will  be  found  on  lifting  the  magnet 
that  the  filings  cling  to  the  ends  in  great  tufts  but  leave 
it  bare  in  the  middle,  as 
shown  in  Fig.  294.  The 
centers  of  attraction  near 
the  ends  are  called  the 

poles  of  the  magnet.     The  FIG.  294.  —  Showing  the  Polarity 

pole  that   points   toward  of  a  Magnet. 

the  north  when  the  bar  is  suspended  is  the  north-seeking, 

or  N-pole,  and  the  other  the  south-seeking,  or  S-pole. 

375.  Law  Of  Magnetic  Poles.  —  Balance  upon  a  pinhead  each  of 
the  magnetized  pieces  of  steel  used  in  §  372  and  mark  the  N-poles 
with  small  labels.  Now  bring  the  N-pole  of  one  piece  near  the  N-pole 
of  a  suspended  one  and  observe  the  effect.  Present  the  N-pole  of  the 
former  to  the  S-pole  of  the  latter.  In  every  case  that  can  be  tested  it 
will  be  found  that  an  N-pole  repels  another  N-pole  and  attracts  an 
S-pole.  Likewise  S-poles  repel  each  other  and  attract  N-poles. 


MAGNETISM 


361 


FIG.  295.  —  A  Horseshoe 
Magnet. 


The  general  law  of  pole  action  is  made  clear  by  this 
experiment,  viz.  like  poles  repel  each  other  and  unlike  poles 
attract. 

The  force  of  attraction  or  repulsion  between  two  poles  is 
inversely  proportional  to  the  square  of  the  distance  between 
them,  i.e.  doubling  the  distance  between  two  poles  divides 
the  force  by  four,  tripling  the  distance  divides  the  force 
by  nine,  etc.  Compare  with  gravitation,  §  67. 

376.  Artificial  Magnets.  —  The  fact  that  artificial  mag- 
nets may  be  made  of  any  desired  form  is  of  great  practical 
value.     The  commonest  forms  are 

the   straight    bar   magnet   and    the 

horseshoe  magnet  shown  in  Fig.  295. 

These  can  be  produced  in  any  size 

from  the  small  toy  magnet  up   to 

large  ones  capable  of  lifting  an  iron   weight   of   several 

pounds. 

Draw  the  N-pole  of  a  magnet  along  a  steel  nail,  from  the  head 
toward  the  point.  Present  the  head  of  the  nail  to  the  N-pole  of  a 
suspended  magnet.  The  observed  repulsion  shows  the  head  to  be 
an  N-pole.  Also  test  the  polarity  of  the  point  of  the  nail.  Repeat 

the  processes,  using  the  S-pole  of  the 
first  magnet,  and  ascertain  the  poles 
produced  in  the  nail. 

In  every  case  it  will  be  found 
that  when  a  bar  is  magnetized  by 
contact  with  one  of  the  poles  of 
a  magnet,  a  pole  of  the  opposite 

name  is  formed  at  the  point  last  touched  by  the  magnet, 

as -illustrated  in  Fig.  296. 

377.  Magnetic  Substances.  —  Practically  only  iron  and 
steel  are  affected  by  a  magnet,  although  the  substances 
nickel  and  cobalt  are  slightly  attracted.     Bismuth,  anti- 
mony, and  some  other  substances  are  appreciably  repelled 


FIG.  296.  —  Magnetizing  a 
Bar  of  Steel. 


362          A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

when  placed  near  a  strong  magnetic  pole.  Bodies  belong- 
ing to  the  former  class  are  called  paramagnetic  or  simply 
magnetic,  substances ;  and  those  of  the  latter,  diamagnetic 
substances. 

378.  Magnetic    Induction Let    a    strong    magnet   support 

an  iron  nail  by  its  head.     Test  the  polarity  of  the  point  of  the  nail. 

Let  the  point  of  the  first  nail  support  'a  second 
one  by  its  head  and  test  the  polarity  of  this 
one  also.  If  the  nails  are  not  too  large,  a 
chain  of  them  may  be  formed  as  in  Fig.  297. 
Now  carefully  remove  the  magnet  from  the 
first  nail,  and  all  will  fall  apart.  Repeat  the 
experiment  after  placing  a  piece  of  paper  be- 
tween the  magnet  and  the  first  nail.  Ascer- 
tain whether  absolute  contact  between  the 
FIG.  297.  — Each  magnet  and  the  nail  is  necessary  in  order  to 

Nail  Becomes    enable  the  first  nail  to  support  the  second. 

a  Magnet.          The  action    of    the    magliet   On   the  nail  is 

simply  weakened  by  the  intervention  of  the 
paper. 

A  piece  of  iron  or  steel  becomes  a  magnet  by  induction 
when  brought  in  contact  with,  or  close  to,  a  magnetic  pole. 
The  effect  takes  place  through  all  substances  except  large 
masses  of  iron  or  steel.  When  the  polarity  of  the  nails 
is  tested,  it  is  found  that  the  N-pole  of  the  magnet,  for 
example,  produces  an  S-pole  on  the  near  end  of  the  nails 
and  an  N-pole  at  the  remote  end.  When  the  magnet  is 
removed,  it  will  be  found  that  a  portion  of  the  induced 
magnetism  is  retained  by  the  nails. 

379.  Retentivity  of  Magnetism. — It  has.  been  evident 
in  many  of  the  preceding  experiments  (§§  372,  376,  378) 
that  a  magnetized  piece  of  steel  loses  only  a  part  of  its 
magnetism  when  it  is  removed  from  the  magnetizing  in- 
fluence.    It  is  almost  impossible  to  find  a  piece  of  iron 
that  will  not  retain  a  little  magnetism  after  being  brought 
in  contact  with  a  magnet,  although  the  amount  retained  is 


MAGNETISM  363 

often  very  slight.  The  property  of  retaining  magnetism  is 
called  retentivity.  Hardened  steel  possesses  the  property 
of  retentivity  in  a  high  degree,  but  in  soft  iron  the  reten- 
tivity is  very  small. 

380.  Magnetic  Fields.  —  A  magnetic  field  is  a  region  in 
which  magnetic  substances  experience  magnetic  forces.  A 
magnetic  field  may  be  represented  in  the  same  manner  as 
an  electric  field  (§  351).  Each  magnetic  line  of  force 
shows  at  every  point  in  it  the  direction  of  the  force  at  that 
point.  Magnetic  fields  are  easily  mapped  by  the  help 
of  iron  filings,  as  in  the  following  experiment : 

Let  a  bar  magnet  be  covered  with  a  sheet  of  paper  and  fine  iron 
filings  strewn  over  its  surface.  If  the  paper  is  slightly  jarred  by  tap- 
ping the  table  on  which  the  magnet  rests,  the  filings  will  arrange 
themselves  in  chains  stretching  from  pole  to  pole.  These  chains 
show  the  direction  of  the  magnetic  forces  at  the  points  through  which 
they  pass. 

The  direction  of  the  lines  of  force  in  the  field  around  a 
short  bar  magnet  is  shown  in  Fig.  298.  Each  particle 
becomes  a  magnet  by  induction  (§  378)  and  turns  until  it 
lies  lengthwise  in  a  line  of  force.  The  poles  of  one  parti- 
cle attract  the  opposite  poles  of  the  neighboring  particles, 
and  thus  the  filings  unite  and  form  chains  which  extend 
from  pole  to  pole. 
The  direction  of  a  line 
of  force  is  assumed  to 
be  from  an  N-pole  to 
an  S-pole  through  the 
air.  The  strong  parts 
of  the  field  are  the  re- 
gions near  the  poles, 
as  shown  by  the 
heavy,  distinct  lines  ' FIQ  ^  _The  ^^  Fwd  ai.ound  . 

of  filings.  Bar  Magnet. 


364 


A  HIGH   SCHOOL   COURSE   IN   PHYSICS 


FIG.  299.  — Showing  the  Magnetic  Field 
between  Unlike  Poles. 


Figure  299  was 
made  by  placing  two 
bar  magnets  side  by 
side  with  their  unlike 
poles  pointing  in  the 
same  direction.  It 
is  at  once  evident  that 
an  intense  field  of 
force  is  produced  be- 
tween unlike  poles 
when  placed  near 
each  other.  This  fact 
is  utilized  in  some  of 
the  practical  applica- 
tions of  magnetism. 

Figure  300  shows 
the  result  obtained  by 
placing  two  short  bar 
magnets  parallel,  but 
with  their  like  poles 
turned  in  the  same 
direction.  A  comparison  of  this  with  the  figure  just  pre- 
ceding shows  at  once  the  presence  of  a  weak  field  of  force 
between  similar  poles.  In  fact,  the  re- 
pellent action  between  the  lines  of  force 
is  obvious.  No  lines  of  force  from  one 
pole  enter  another  pole  of  the  same 
name.  Since  the  poles  are  alike,  it  is 
plain  that  this  field  represents  the  case 
of  repulsion,  while  Fig.  299  shows  the 
condition  for  attraction. 

381.,  Magnetic  Permeability.  —  Mag- 
netic lines  of  force  find  an  easier  path 
through  iron  or  steel  than  through  air. 


FIG.  200.  —  Showing  the  Magnetic  Field 
between  Like  Poles. 


FIG.  301.  —  A  Magnetic 
Field  Distorted  by  a 
Piece  of  Iron  A. 


MAGNETISM  365 

When  a  piece  of  iron,  A,  Fig.  301,  is  placed  in  a  magnetic 
field,  lines  of  force  bend  from  their  original  course  in 
order  to  pass  through  the  iron.  Thus  lines  of  force  are 
sent  through  the  metal.  The  relative  ease  with  which  mag- 
netic lines  of  force  pass  through  a  substance  is  called  its  mag- 
netic permeability.  The  permeability  of  air  is  regarded  as 
unity. 

EXERCISES 

1.  What  would  be  a  suitable  substance  for  permanent  artificial 
magnets  ? 

2.  It  is  desirable  in  a  certain  instrument  to  use  a  substance  that  is 
easily  magnetized,  but  which  will  lose  its  magnetism  when  the  mag- 
netizing influence  is  removed.     What  would  be  the  proper  substance  ? 

3.  Each  of  two  nails  hangs  from  the  N-pole  of  a  permanent  mag- 
net.    Will  the  ends  of  the  nails  remote  from  the  magnet  attract  or 
repel  each  other  ? 

4.  Four  nails  are  placed  lengthwise  in  a  row  without  touching  each 
other.     Draw  a  figure  showing  the  magnetic  fields  produced  when  the 
S-pole  of  a  magnet  is  placed  near  one  end  of  the  row. 

5.  The  N-pole  of  a  magnet  is  held  near  a  point  on  the  circumfer- 
ence of  an  iron  ring.     Show  by  a  diagram  the  position  of  the  induced 
magnetic  poles. 

2.   MAGNETISM  A  MOLECULAR  PHENOMENON 

382.  Demagnetization  by  Heating. —Magnetize  a  piece  of 
watch  spring  and  note  the" quantity  of  iron  filings  that  it  will  support. 
Heat  the  spring  red-hot  and  test  its  magnetism  again.     It  will  be  found 
to  have  lost  its  power  of  picking  up  filings  as  well  as  repelling  either 
pole  of  a  suspended  magnet. 

We  have  already  learned  in  §  213  that  heating  a  body 
simply  increases  its  molecular  motion.  This  experiment 
shows,  therefore,  that  when  the  molecular  motion  reaches 
a  certain  degree,  practically  all  magnetism  is  destroyed. 

383.  Demagnetization  by  Molecular  Rearrangement  — 

Bend  a  piece  of  iron  wire  in  the  form  shown  in  Fig.  302.  Magnetize 
it  by  stroking  it  several  times  with  a  strong  magnet.  Test  its  power 


366          A  HIGH   SCHOOL   COURSE   IN   PHYSICS 

to  pick  up  filings  and  to  repel  the  pole  of  a  suspended  magnet.     Now 
grasp  the  ends  with  pliers  and  give  the  wire  a  vigorous  twist.     If  the 
wire  is  now  tested  as  before,  it  will  be  found 
"      to  have  lost  its  magnetism. 
FIG.  302.  —  Wire  may 

be  Demagnetized  by          It   is   obviOUS  that  the  effect  of  twist- 

ing  is  to  give  all  parts  of  the  wire  a  new 
molecular  arrangement.  Accompanying  this  disarrange- 
ment is  the  disappearance  of  the  magnetism,  as  shown  by 
the  experiment. 

384.   Magnetism  Not  Simply  at  the  Poles.  — -  Magnetize  a 

piece  of  watch  spring  about  5  in.  long  and  test  the  location  of  its  poles 
by  presenting  it  to  a  suspended  magnet.  Break  it  at  the  center  and 
test  the  pieces.  It  will  be  found  that  each  piece  is  a  perfect  magnet, 
for  two  new  poles  will  have  developed  at  the  point  which  was  at  first 
neutral.  Break  one  of  the  pieces  at  its  center  and  again  each  piece 
will  be  a  perfect  magnet. 

Let  the  original  magnet  be  represented  by  (1),  Fig.  303. 
The  polarity  manifests  itself  only  at  the  ends.  When  the 
magnet  is  broken,  as 

at  P,  the   condition        JS^jjij?  :g|  ( / ) 

shown   in  (2)  is  the 
result.       Again,     on 


breaking      the      two     «-     *  " 


_  . P(3) 

parts,     the     result         FIG.  303. -Effect  of  Breaking  a  Magnet. 

shown  in  (3)  is  ob- 
tained.    It  may  be  imagined  that  this  process  be  carried  on 
even  to  the  separation  of  ultimate  particles,  i.e.  the  mole- 
cules.    There  can  be  no  doubt  that  each  molecule  would 
prove  to  be  a  magnet  having  two  poles. 

385.  Theory  of  Magnetization. — The  results  obtained 
in  the  experiments  of  the  preceding  sections  lead  to  the 
theory  of  magnetization  which  assumes  that  every  molecule 
of  iron  or  steel  is  a  magnet  even  when  the  bar  of  which  it  is  a 
part  is  not  magnetized.  Magnetization  consists  in  causing 


MAGNETISM 


367' 


FIG.  304.  —  Illustrating  the  Condition  in  an  Un- 
magnetized  Bar  of  Iron  or  Steel. 


FIG.  305. 


Illustrating  the  Condition  in  a 
Magnetized  Bar. 


the  molecules  to  arrange  themselves  in  a  certain  order. 
In  an  unmagnetized  bar  of  iron  the  condition  is  repre- 
sented as  in  Fig.  304.  Here  each  of  the  little  rectangles 
represents  a  pivoted  magnet,  the  N-pole  being  shaded.  It 
will  be  seen  that  the  small  magnets  arrange  themselves  in 
small  groups  so  that 
unlike  poles  neu- 
tralize each  other 
throughout  the  bar. 
The  bar,  therefore, 
manifests  no  polar- 
ity. 

When  a  magnet- 
izing influence  is 
brought  to  bear 
upon  the  bar  of  iron, 
the  molecules  swing 
round  until  a  condi- 
tion approximating 
that  shown  in  Fig. 
305  results.  Since 

.,  -IT.        , .  FIG.  306.  —  Illustrating  the  Condition  in  a 

the  general  direction  Saturated  Magnet, 

of  the  N-poles  is  to- 
ward the  left  and  the  S-poles  toward  the  right,  the  ends  of 
the  bar  will  show  polarity.  At  a  short  distance  from  the 
ends  and  throughout  the  middle  of  the  bar  the  proximity 
of  unlike  poles  brings  about  a  neutralization  of  polarity  in 
these  places.  A  jar  serves  to  break  up  this  artificial  ar- 
rangement, which  thereby  destroys  the  magnetism,  and  the 
condition  shown  in  Fig.  304  is  resumed. 

386.  A  Saturated  Magnet.  —  If  the  theory  of  magnetiza- 
tion just  described  is  correct,  it  will  be  found  impossible 
to  magnetize  a  piece  of  iron  beyond  the  limit  reached 
when  all  the  molecules  have  been  turned  as  represented 


una  ma  ma  mna  ma  ma  na  ma  ran  ma 
ma  HDD  na  ID  ID  oa  ma 


368          A  HIGH   SCHOOL   COURSE   IN   PHYSICS 

in  Fig.  306.  This  has  been  found  experimentally  to  be 
the  case.  In  this  condition  a  magnet  is  said  to  be  satu- 
rated. Hence,  a  saturated  magnet  is  one  upon  which  an 
increase  of  the  magnetizing  influence  has  no  effect. 

EXERCISES 

1.  How  would  you  determine  the  poles  of  a  magnet?    Give  two 
methods. 

2.  Why  is  a  permanent  magnet  injured  when  dropped? 

3.  If  iron  be  heated  and  then  cooled  in  a  magnetic  field,  it  will  be 
found  to  be  magnetized.     Explain. 

4.  Explain  how  jarring  a  bar  of  steel  will  aid  in  magnetizing  it, 
but  jarring  a  magnetized  piece  of  steel  will  weaken  its  magnetism. 

3.   TERRESTRIAL  MAGNETISM 

387.  The  Compass.  —  One  of  the  earliest  properties  dis- 
covered regarding  a  magnet  is  its  tendency  to  take  a  defi- 
nite position  when  placed  on  a  pivot  or  suspended.  A 
magnet  that  is  so  pivoted  as  to  turn  freely  in  a  horizontal 
plane  is  called  a  compass.  The  invention  of  the  compass 
is  attributed  to  the  Chinese. 

The  first  satisfactory  explanation  of  the  action  of  the 
compass  was  given  by  Gilbert1  about  1600  A.D.  The 

1  WILLIAM  GILBERT  (1540-1603).  The  renown  of  Gilbert  rests  largely 
upon  the  fact  that  he  was  one  of  the  first  to  recognize  the  value  of  experi- 
mentation and  also  upon  his  great  work  entitled  De  Magnete,  which  was 
published  in  London  in  1600.  His  most  celebrated  experiments  were 
made  with  magnets  and  magnetic  bodies,  and  his  results  and  conclusions 
are  contained  in  the  book  just  named.  Gilbert  made  the  discovery  that 
the  earth  is  a  great  magnet  and  demonstrated  this  by  constructing  a 
small  sphere  of  lodestone.  With  this  "terrella,"  or  "little  earth,"  he 
was  not  only  able  to  show  why  magnets  point  to  the  north,  but  he  also 
explained  the  declination  and  inclination  of  magnetic  needles. 

Gilbert  also  made  important  discoveries  in  the  subject  of  Electricity. 
He  was  probably  the  first  to  clearly  recognize  a  distinct  difference  between 
magnetized  and  electrified  bodies.  He  showed  that  many  substances  be- 
sides amber  could  be  electrified  by  rubbing  ;  e.g.  glass,  resins,  sealing  wax, 


MAGNETISM  369 

assumption  is  made  that  the  earth  is  a  great  magnet,  but 
the  reason  for  its  being  one  still  remains  a  mystery. 
However,  we  are  sure  that  the  earth  is  surrounded  by  a 
magnetic  field,  and,  in  proof  of  this  fact,  the  following 
experiment  can  easily  be  made. 

Hold  a  bar  of  iron  or  a  slender  gas  pipe  about  2  feet  long  in  a  north- 
arid-south  plane,  tilting  the  north  end  down  60°  or  more.  Now  give 
the  bar  several  vigorous  taps  with  a  stone  and  then  test  the  ends  for 
polarity  by  presenting  them  to  the  poles  of  a  suspended  magnet.  The 
end  of  the  bar  toward  the  north  will  be  an  N-pole,  and  the  other  an 
S-pole.  Reverse  the  bar  and  repeat  the  processes.  The  polarity  of 
the  bar  will  be  found  reversed. 

This  experiment  makes  use  of  the  earth's  magnetic  field 
in  the  magnetization  of  the  iron  bar.  The  jarring  is  ef- 
fective only  in  lending  assistance  to  the  rearrangement  of 
the  molecules  under  the  inductive  influence  of  the  earth's 
field  (§§  380,  385).  The  compass  needle  simply  behaves 
as  any  small  magnet  would  in  a  magnetic  field,  pointing 
in  a  general  north-and-south  direction  because  of  the 
influence  of  the  earth's  lines  of  force. 

388.  Declination  of  the  Needle.  —  Suspend  a  magnetized  knit- 
ting needle  on  a  fiber  taken  from  the  cocoon  of  a  silkworm,  or  on  an 
untwisted  filament  of  silk  floss.  Stretch  a  cord  below  the  needle  pre- 
cisely in  a  north-and-south  line  (given  by  the  shadow  of  a  plumb  line 
or  a  vertical  window  frame  at  noon,  sun  time).  The  experiment 
should  not  be  made  within  several  feet  of  an  iron  pipe  or  beam,  nor 
within  a  foot  of  steel  nails.  It  will  be  observed  that  the  needle  does 
not  take  a  position  parallel  to  the  north-and-south  line  except  in 
certain  localities. 

etc.  These  he  named  "  electrics,'1  from  the  Greek  word  electron,  meaning 
amber.  He  was  also  the  first  to  use  the  word  "  electricity." 

Gilbert  was  educated   in  medicine  at  Cambridge,   England,  and  was 
appointed  court  physician  by  Queen  Elizabeth.     At  the  death  of  the  queen 
in  1603,  he  was  reappointed  by  her  successor,  James  I,   but  his  death 
occurred  in  November  of  that  year. 
25 


370 


A  HIGH   SCHOOL   COURSE   IN   PHYSICS 


The  earth's  magnetic  lines  do  not  coincide  with  the  geo- 
graphical meridians ;  consequently  the  magnetic  needle, 
directed  by  the  earth's  field,  will  not  point  geographically 
north.  This  fact  was  known  as  early  as  the  eleventh 
century ;  but  it  was  first  discovered  by  Columbus,  on  his 
memorable  voyage  of  1492,  that  the  direction  indicated 
by  the  compass  changes  as  one  passes  from  place  to  place 
over  the  earth's  surface.  The  angle  between  the  direction 
of  the  needle  and  the  geographical  meridian  is  the  declina- 
tion of  the  needle. 

Lines  that  are  so  drawn  upon  a  map  as  to  pass  through 
places  at  which  the  declination  is  the  same  are  called 


FIG.  307. — Map  Showing  Magnetic  Declinations.  Lines  of  no  Declination 
are  Marked  "0."  The  Numbers  Show  the  Declination  in  Degrees  and  the 
Letters  Show  Whether  it  is  East  or  West. 

isogonic  lines.  Figure  307  shows  approximately  the  mag- 
netic declination  at  all  places  on  the  surface  of  the  earth. 
The  heavy  line,  called  the  agonic  line,  shows  the  regions 
where  the  needle  points  due  north.  At  all  points  in  the 
United  States  and  Canada  lying  east  of  the  agonic  line, 


MAGNETISM  371 

the  declination  is  towards  the  west ;  for  all  points  west 
of  this  line,  the  declination  is  east.  At  the  present  time 
(1910)  the  agonic  line  passes  a  little  west  of  Lansing,  Mich., 
and  slightly  east  of  Cincinnati,  Ohio,  and  Charleston,  S.C. 
It  is  moving  very  slowly  westward. 

389.  Inclination  of  the  Magnetic  Needle.— Thrust  an  unmag- 

netized  knitting  needle  through  a  cork,  and  close  to  and  at  right  angles 
with  it  pass  a  straight,  slender  sewing  needle.  Cut  away  a  portion 
of  the  cork  until  the  system  will  balance  in 
any  position  on  the  edges  of  two  tumblers 
when  the  longer  needle  is  placed  east-and- 
west.  Magnetize  the  knitting  needle  by  strok- 
ing one  end  with  the  N-pole  of  a  magnet  and 
the  opposite  end  with  the  S-pole.  Now  place 
the  system  in  a  north-and-south  position  on 
the  tumblers.  The  N-pole  of  the  needle  will 
appear  heavier  than  the  S-pole  and  will  dip 
until  the  angle  between  the  needle  and  the 
horizontal  plane  is  about  70°.  (See  Fig.  308.)  FlG  3Q8  _  The  N_pole 

The  balanced,  or  dipping,  needle 
simply  places  itself  parallel  with  the 
lines  of  force  of  the  earth.  In  the  Northern  States  these 
lines  are  inclined  about  70°  from  the  horizontal.  The 
angle  between  the  earth's  magnetic  lines  of  force  and  a  hori- 
zontal plane  is  called  the  inclination,  or  dip,  of  the  needle. 
The  angle  of  inclination  increases  as  one  approaches  the 
magnetic  poles  of  the  earth,  where  it  is  90°.  Near  the 
geographical  equator  the  inclination  is  about  0°,  and  in 
the  southern  hemisphere  the  needle  inclines  with  its  S-pole 
below  the  horizontal  plane  passing  through  its  axis. 

EXERCISES 

1.  Account  for  the  fact  that  the  iron  beams  of  a  building,  gas  and 
water  pipes,  and  other  bars  of  iron  are  usually  magnetized. 

2.  How  would  a  compass  behave  while  being  carried  entirely 
around  the  earth's  magnetic  pole  along  the  Arctic  Circle  ? 


372          A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

3.  How  would  a  dipping  needle  be  of  assistance  in  locating  the 
magnetic  poles  of  the  earth  ? 

4.  Account  for  the  fact  that  a  surveyor's  compass  needle  is  often 
provided  with  a  small  adjustable  weight  for  balancing. 

5.  Place  a  ruler  upon  the  table,  giving  it  approximately  the  declina- 
tion of  the  needle  at  New  York  ;  at  Los  Angeles ;  at  Iceland. 

6.  Ascertain  by  experiment  whether  a  floating  magnet  tends  to 
drift  toward  the  north.     Account  for  the  result. 

SUGGESTION.  —  Consider  whether  the  attraction  of  one  of  the 
earth's  poles  for  a  pole  of  a  magnet  is  greater  than  its  repulsion  for 
the  opposite  pole  of  the  same  magnet. 

7.  The  upper  end  of  a  pipe  driven  into  the  ground  was  found  to  be 
an  S-pole.     Explain.     Suggest  an  experiment  by  which  your  explana- 
tion can  be  tested. 

SUMMARY 

1.  A  bar  of  steel  may  be  made  a  magnet  by  discharging 
a  Ley  den  jar  through  a  helix  of  wire  wound  around  it 
(§  372). 

2.  Magnets  also    appear  in  nature  in  an  ore  of  iron 
called  magnetite.     Specimens  of  this  ore  that  are  magnets 
are  called  natural  magnets  (§  373). 

3.  On  account  of  the  tendency  of  a  magnet  to  place 
itself  in  a  north-and-south  position,  one  end  is  called  the 
north-seeking,  or  N-pole ;  the  other,  the  south-seeking,  or 
S-pole  (§  374). 

4.  Poles    of    the    same    name    repel    each    other,    and 
those  of  unlike  name  attract  (§  375). 

5.  Artificial   magnets  may  be  made  by  stroking  bars 
of  steel  with  a  magnetized  body  (§  376). 

6.  Substances  ^that  are  attracted  by  a  magnet  are  called 
paramagnetic  (or  simply  magnetic)  substances ;  those  that 
are  repelled,  diamagnetic.     Iron  is  the  most  magnetic  of 
all  substances  (§  377). 

7.  A  piece  of  iron  or  steel  becomes  a  magnet  by  indue- 


MAGNETISM  373 

tion  when  brought  near  the  pole  of  a  magnet.  An  N-pole 
always  induces  an  S-pole  on  the  nearer  end  of  a  neighbor- 
ing bar  of  iron  or  steel,  and  an  S-pole  induces  an  N-pole 
(§  378). 

8.  The  region  around  a  magnet   in  which  magnetic 
substances  experience  magnetic  forces  is  called  a  magnetic 
field  (§  380). 

9.  Magnetic  lines  of  force  pass  more  readily  through 
iron  than  air.     Different  grades  of  iron  and  steel  differ 
also  in  the  ease  with  which  lines  of  force  pass  through 
them.     They  are  therefore  said  to  differ  in  permeability 
(§  381). 

10.  The  magnetism  which  would  ordinarily  be  retained 
by  iron  or  steel  may  be  reduced  or  entirely  destroyed  by 
heating,  jarring,  twisting,  etc.     Hence  magnetism  is  con- 
sidered to  be  a  molecular  phenomenon.     The  molecules  of 
iron  and  steel  are  supposed  to  be  small  magnets  having  two 
poles.     Magnetization  consists  in  giving  the  molecules  a 
new  arrangement  in  which  the  N-poles  point  in  general  in 
one  direction  and  the  S-poles  in  the  other  (§§  382  to  386). 

11.  A  magnet  so 'pivoted  as  to  turn  freely  in  a  horizontal 
plane  is  called  a  compass.     It  takes  a  general  north-and- 
south  position  on  account  of  the  earth's  magnetism  (§  387). 

12.  The  angle  that  measures  the  deviation  of  the  compass 
from  the  geographical  meridian  is  called  the  declination  of 
the  needle.     This  angle  varies  greatly  with  different  lo- 
calities.    The  agonic  line  is  the  line  passing  through  points 
where  the  declination  is  zero.      It  is  now  moving  slowly 
westward  (§  388). 

13.  The  angle  between  the  earth's  magnetic  lines  and  a 
horizontal  plane  is   called  the   inclination,  or   dip,  of  the 
needle.     The  dip  varies  from  zero  at  the  magnetic  equator 
to  90°  at  the  earth's  magnetic  poles  (§  389). 


CHAPTER   XVIII 


VOLTAIC   ELECTRICITY 

1.   PRODUCTION   OF   A   CURRENT  —  VOLTAIC   CELLS 

390.  Maintaining  a  Continuous  Discharge  of  Electricity. 
—  It  was  shown  in  §  372  that  the  moving  charge  of  elec- 
tricity obtained  when  a  Leyden  jar  is  discharged  through 
a  helix  of  wire  is  able  to  magnetize  pieces  of  steel.  But 
electricity  would  be  of  very  little  service  if  we  were  obliged 
to  depend  upon  the  momentary  flow  occasioned  by  such  a 
discharge.  The  great  practical  value  of  electricity  as  a 
working  agent  lies  in  the  fact  that  it  is  possible  by  differ- 
ent means,  now  to  be  studied,  to  maintain  a  continuous  flow 
of  electricity,  or  in  other  words,  a  steady  current.  The 
following  experiments  afford  an  opportunity  to  compare 
these  two  kinds  of  discharge,  —  viz.  the  momentary  and 
the  continuous. 

1.  Place  one  end  of  a  coil  of  insulated  wire  within  which  are  several 
pieces  of  soft  annealed  iron  wire  near  one  of  the  poles  of  a  light 

pivoted  magnetic  needle  as 
shown  in  Fig.  309.  Discharge 
a  highly  charged  Leyden  jar 
through  the  wire  of  the  coil 
and  observe  the  effect  upon 
the  needle.  The  pole  will  be 
either  attracted  or  %  repelled. 
Except  for  the  effect  of  the 
magnetism  remaining  in  the 
iron,  the  action  is  only  tempo- 
rary. Recharge  the  jar  and 


FIG.  309.  —  Magnetizing    Iron    by   a  Mo- 
mentary Discharge  of  Electricity. 

374 


discharge  it  through  the  coil 


VOLTAIC   ELECTRICITY  375 

in  the  opposite  direction.     If  the  first  discharge  produced  repulsion, 
this  one  will  set  up  an  attraction. 

2.  Attach  one  end  of  the  coil  shown  in  Fig.  310  to  a  strip  of  copper 
about  10  cm.  long  and  the  other  to  a  strip  of  zinc.  Now  dip  both 
plates  in  a  very  dilute  solu- 
tion (about  1 : 40,  by  volume) 
of  sulphuric  acid,  keeping 
the  plates  from  touching 
each  other.  One  of  the  mag-  N 
netic  poles  of  the  pivoted 
needle  will  swing  round 
toward  the  coil  and  will  not 
resume  its  original  position 
so  long  as  the  plates  remain 
in  the  liquid.  Reverse  the  FIG.  310. -Magnetizing  Iron  by  a  Continuous 

,.  ,        ,  Current, 

connections  at  the  ends  of 

the   coil,   and  the  opposite  end  of    the   needle   will  be   attracted. 

These  experiments  show  a  marked  similarity  between 
the  effect  produced  by  the  discharge  of  a  Leyden  jar  and 
that  produced  by  the  plates  of  zinc  and  copper  suspended 
in  a  solution  of  sulphuric  acid.  The  iron  is  magnetized 
in  both  instances,  and  the  magnetism  is  even  reversed  by 
reversing  the  connections.  But  since  the  effect  in  Ex- 
periment 2  lasts  as  long  as  both  plates  remain  in  the 
liquid,  it  is  clear  that  a  continuous  discharge  can  be  main- 
tained by  the  means  employed.  A.  continuous  discharge, 
or  movement,  of  electricity  is  called  an  electric  current. 

391.  The  Voltaic  Cell. — The  combination  of  zinc,  cop- 
per, and  dilute  sulphuric  acid  used  in  Experiment  2  of  the 
preceding  section  constitutes  a  voltaic  cell.  This  method 
of  producing  a  current  was  first  used  by  Volta1  in 
the  year  1800 ;  hence  the  name.  There  are  many  kinds 
of  voltaic  cells,  differing  from  one  another  in  respect  to 
the  materials  used  in  their  construction.  Some  of  the 
most  important  ones  are  treated  later  on. 

1  See  portrait  facing  page  352. 


376         A  HIGH   SCHOOL   COURSE   IN   PHYSICS 

392.  Electrical    Charges  produced    by  Voltaic  Cells.— 

Provide  two  perfectly  flat  metal  disks  about  3  in.  in  diameter.  Attach 
one  disk  to  the  top  of  a  sensitive  gold-leaf  electroscope  and  give  it  a  thin 
coating  of  shellac.  To  the  second  disk  attach  an  insulated  handle 
and  place  it  upon  the  first.  The  two  plates  thus  form  a  condenser  of 
large  capacity.  Why?  See  §365.  Form  a  series1  of  voltaic  cells 
(discarded  dry  cells  will  usually  do)  and  touch  the  wire  leading  from 

the  zinc  plate  to  the  upper  disk  A, 
Fig.  311,  and  the  wire  leading  from 
the  carbon  (or  copper)  to  disk  B. 
Remove  the  wires  and  lift  the 
upper  disk.  The  leaves  of  the 
electroscope  will  diverge.  On  bring- 
ing an  electrified  stick  of  sealing 
wax  near  the  instrument,  the  diver- 
gence will  be  decreased,  thus  show- 
ing that  the  carbon  (or  copper) 
FIG.  311. -Charging  an  Electroscope  late  conveyed  to  the  lower  disk  a 
from  Voltaic  Cells. 

positive  charge.     By  repeating  the 

experiment  with  the  wires  reversed,  it  will  be  found  that  a  negative 
charge  can  be  taken  from  the  zinc  of  the  cell. 

It  becomes  clear  from  this  experiment  that  the  terminals 
of  a  voltaic  cell  and  the  wires  leading  from  them  are  electri- 
cally charged^  —  a  positive  charge  being  borne  by  the  carbon 
(or  copper)  and  a  negative  charge  by  the  zinc.  Hence  the 
copper  of  a  cell  is  called  the  positive  pole,  and  the  zinc,  the 
negative  pole.  The  voltaic  cell,  therefore,  affords  a  case  of 
two  oppositely  electrified  bodies,  which  on  being  joined  by 
a  conductor  set  up  an  electrical  discharge.  Furthermore, 
this  discharge,  or  ^current,  is  continuous;  for,  as  fast  as 
the  charges  become  neutralized,  they  are  renewed  by  the 
chemical  changes  taking  place  within  the  cell. 

393.  Electromotive  Force. — As  shown  in  the  preceding 
section,  a  voltaic  cell  develops  a  charge  of  positive  electri- 
fication on  the  copper  plate  and  a  charge*  of  negative  on 

1  If  the  disks  are  perfectly  flat  and  the  insulating  material  between 
them  is  thin  enough,  the  experiment  can  be  made  by  using  a  single  cell. 


VOLTAIC   ELECTRICITY  377 

the  zinc.  The  difference  of  potential  that  the  cell  maintains 
between  these  two  charges  when  the  plates  are  not  connected  by 
a  conductor  is  the  electromotive  force  (abbreviated  E.  M.  F.) 
of  the  cell.  E.  M.  F.  is  sometimes  called  electric  pressure 
and  is  the  cause  of  an  electric  flow  in  a  circuit.  It  is  not 
a  force  in  the  sense  in  which  that  term  is  used  in  mechanics, 
since  its  tendency  is  to  move  electricity  and  not  matter. 
The  unit  of  E.  M.  F.  is  the  volt,  which  is  approximately 
the  E.  M.  F.  afforded  by  a  cell  containing  zinc,  copper,  and 
dilute  sulphuric  acid. 

394.  An  Electrical  Circuit.  —  The  entire  conducting  path 
along  which  a  current  of  electricity  flows  is  called  an  elec- 
trical circuit.     The  circuit   comprises  not  only  the  wire 
with  which  the  plates  are  connected,  but  also  the  plates 
and  the  liquid  of  the  cell.     When  an  instrument,  for  ex- 
ample, is  to  be  introduced  into  the  circuit,  it  is  so  connected 
with  the  plates  of  the  cell  as  to  be  traversed  by  the  cur- 
rent and  is  thus  made  a  part  of  the  circuit.     Separating 
the  circuit  at  any  point  is  called  opening,  or  breaking,  the 
circuit ;  and  joining  the  separated  ends  is  called  closing,  or 
making,  the  circuit.     When  a  circuit  is  broken,  no  current 
can  flow,  and  the  circuit  is  therefore  said  to  be  interrupted. 

395.  Action  of  a  Voltaic  Cell.  —  1.    Place  a  strip  of  commer- 
cial zinc  in  very  dilute  sulphuric  acid  (about  1  : 40,  by  volume)  and 
observe  the  effect.     Very  small  bubbles  will  be  seen  to  rise  from  the 
zinc  and  pass  off  at  the  surface  of  the  liquid.     These  are  bubbles  of 
hydrogen  gas  which  is  liberated  from  the  acid  by  the  chemical  action. 
If  a  small  piece  of  zinc  be  left  in  the  acid,  it  will  soon  dissolve,  leaving 
behind  only  a  few  small  flakes  of  insoluble  impurities.     Repeat  the 
experiment  with  the  strip  of  copper.     No  action  will  be  observed. 

2.  Touch  the  zinc  plate  just  used  to  a  small  quantity  of  mercury. 
Some  of  the  mercury  will  be  found  to  cling  to  the  plate.  With  a  cloth 
or  sponge  spread  the  mercury  over  the  wet  surface  of  the  zinc.  Now 
on  placing  the  zinc  in  the  acid,  no  bubbles  will  be  seen.  Place  the 
copper  plate  in  the  acid  with  the  zinc,  but  not  in  contact  with  it,  and 
connect  the  two  rnetals  by  means  of  a  wire.  Bubbles  will  now  be 


378 


A  HIGH   SCHOOL   COURSE   IN   PHYSICS 


observed  rising  from  the  copper  plate.  It  is  under  these  conditions, 
as  we  have  seen  in  §  390,  that  a  magnetic  effect  is  derived  from  the 
wire  which  connects  the  two  plates.  If  the  circuit  be  now  broken  at 
any  point,  the  bubbles  cease  to  form.  The  mercury  has  thus  pre- 
vented the  formation  of  hydrogen  bubbles  at  the  zinc  plate,  which, 
nevertheless,  continues  to  dissolve  and  decrease  in  size  as  long  as  the 
electrical  connection  is  maintained  between  it  and  the  copper. 

It  is  clear  from  Experiment  1  that  the  acid  used  acts 
very  unequally  upon  the  two  plates  of  a  voltaic  cell. 
It  is  this  difference  in  the  chemical  action  that  gives 
rise  to  the  difference  of  potential  between  the  positive  and 
negative  charges  found  in  §  392.  The  greater  the  disparity 
in  the  chemical  actions  at  the  two  plates,  the  greater  the  dif- 
ference of  potential  maintained  by  the  cell.  Furthermore, 
the  experiment  shows  that  the  negatively  charged  plate 
(i.e.  the  zinc)  is  dissolved  by  the  acid,  while  the  posi- 
tive copper  plate  from  which  hydrogen  rises  during  the 
operation  of  the  cell  remains  apparently  unchanged. 


Zinc 


Copper 


©00© 


396.  .Theory  of  a  Voltaic  Cell.  —  The  theory  of  the  simple 
voltaic  cell  described  in  §  390  rests  upon  the  hypothesis  of  Clausius, 

(1822-1888),  a  German  physicist.  This 
hypothesis,  which  is  based  upon  a  large 
amount  of  experimental  evidence, 
states  that  many  of  the  molecules  in 
a  dilute  solution  of  a  substance  "  split 
up  "  into  two  parts  called  ions.  Hence, 
when  sulphuric  acid  (HgSO^1  is  di- 
luted, two  kinds  of  ions  are  formed, 
those  of  hydrogen  (H)  and  those  of 
SO4,  called  the  sulphions.  See  Fig. 
312.  Furthermore,  the  hydrogen  ions 
bear  positive  charges  of  electricity,  while  the  sulphions  carry  negative 
charges.  As  shown  in  §  395,  zinc  has  a  strong  tendency  to  dis- 
solve in  dilute  sulphuric  acid,  while  copper  has  not.  In  the  process 

1  This  chemical  formula  expresses  the  fact  that  each  molecule  of 
sulphuric  acid  is  composed  of  two  atoms  of  hydrogen,  one  of  sulphur, 
and  four  of  oxygen. 


FIG.    312.  —  Diagram     Showing 
Ions  in  a  Voltaic  Cell. 


VOLTAIC   ELECTRICITY 


379 


the  sulphions  in  the  liquid  attack  the  zinc  plate,  from  which  they 
abstract  some  of  the  metal  to  form  zinc  sulphate  (ZnSO4),  a  white 
substance  that  dissolves  at  once  in  the  liquid.  In  this  process  the 
negative  charge  carried  by  each  sulphion  concerned  in  the  action  is 
given  up  to  the  zinc  plate,  which  thus  becomes  charged  with  negative 
electricity.  Again,  for  each  negatively  charged  ion  that  engages  with 
zinc,  a  positive  hydrogen  ion  from  the  liquid  gives  up  its  charge  to 
the  copper,  thus  charging  it  with  positive  electricity.  After  the  hy- 
drogen ions  have  discharged  their  electricity  to  the  copper  plate,  they 
become  free  hydrogen,  which  collects  in  small  bubbles  at  this  plate 
and  rises  to  the  surface. 

397.  Local  Action  and  its  Prevention.  —  The  results  of 
Experiment  2   (§   395)  show  that  a  coating  of  mercury 
upon  the  zinc  plate  prevents  the  formation  of  hydrogen  at 
its  surface  when  it  comes  in  contact  with  the  acid.     The 
reason  is  because  pure  zinc  will  not  dis- 
solve in  pure  sulphuric  acid ;    and  the 

mercury  dissolves  from  the  plate  only  the 
pure  zinc  which  is  then  coated  over  the 
surface,  thus  overlaying  the  impurities 
with  an  amalgam  of  zinc.  After  a  time 
these  impurities,  which  are  mainly  carbon 
and  iron,  become  exposed  to  the  acid, 
and  local  currents  are  set  up  between 
them  and  the  neighboring  portions  of  the 
plate,  as  indicated  by  the  arrows  in  Fig.  313.  The  gen- 
eration of  electric  currents  between  the  zinc  of  a  cell  and  its 
impurities  is  called  local  action.  Local  action  is  a  wasteful 
process,  but  can  obviously  be  prevented  by  employing 
pure  zinc  or  by  coating  an  impure  zinc  plate  with  mer- 
cury. This  treatment  is  known  as  the  amalgamation  of 
the  zinc  plate. 

398.  A  Mechanical  Analogy.  —  The  mechanical  device 
shown  in  Fig.   314  is  of  assistance  in  making  clear  the 
action  of  a  voltaic  cell.     Imagine  a  rotary  pump  P  to  be 


FIG.  313.  — Wasting 
of  the  Zinc  by 
Local  Action. 


380 


A  HIGH   SCHOOL   COURSE   IN   PHYSICS 


FIG.  314.  — A  Mechan- 
ical Analogy  of  Cell 
Action. 


placed  in  a  U-fcube  and  arranged  to  be  turned  by  the 
weight  W.  When  the  wheel  of  the  pump  is  turned,  water 
is  forced  to  a  greater  height  in  one  arm 
than  in  the  other.  The  wheel,  how- 
ever, will  come  to  rest  when  the  back 
pressure  due  to  the  difference  of  level 
on  the  two  sides  of  it  just  equals  the 
pressure  exerted  by  the  wheel.  If  there 
is  no  friction,  the  device  will  maintain 
the  difference  in  level  h  as  long  as  no 
water  escapes. 

In  this  system  the  difference  of  level 
h  maintained  by  the  pump  is  analogous 
to  the  difference  of  potential  (E.  M.  F.)  maintained  be- 
tween the  plates  of  a  voltaic  cell.  Just  as  the  pump 
ceases  to  turn  when  a  certain  difference  of  level  is  reached, 
so  the  chemical  action  of  a  perfect  cell  stops  when  the 
difference  of  potential  between  the  positive  and  negative 
charges  on  the  plates  has  attained  a  certain  value,  which 
will  depend  on  the  nature  of  the  ma- 
terials used  in  its  construction. 

399.  A  Cell  in  Action. — The  case 
of  a  voltaic  cell  is  modified  as  soon 
as  the  circuit  is  completed  and  a  cur- 
rent allowed  to  flow ;  so  also  is  that 
of  the  pump  and  water.  Imagine  a 
pipe  T7,  Fig.  315,  to  connect  the  two 
arms  of  the  U-tube.  On  account  of 
the  difference  of  water  level,  the  liquid 
will  flow  through  T,  and  the  wheel  will 
continue  to  turn,  since  now  the  back 
pressure  will  have  been  reduced.  The 
difference  of  level  will  decrease  to  h',  whose  value  will 
depend  on  the  friction  offered  to  the  current  flow  in  T. 


FIG.  315.  — When  Water 
is  Allowed  to  Flow 
through  T,  the  Wheel 
at  P  will  Continue  to 
Turn. 


VOLTAIC   ELECTRICITY  381 

Similarly,  when  the  plates  of  a  voltaic  cell  are  connected 
by  a  conductor,  the  charge  on  the  positive  plate  (copper) 
moves  toward  the  negative  plate  (zinc),  and  the  potential 
difference  is  diminished.  The  chemical  action  now  goes 
on  vigorously  in  its  attempt  to  restore  the  charge  that 
has  passed  through  the  conductor.  Hence  the  difference 
of  potential  between  the  two  poles  of  a  cell  when  a  current 
is  flowing  will  be  less  than  that  when  the  circuit  is  open. 
This  value  is  no  longer  called  the  E.  M.  F.  of  the  cell,  but 
is  termed  the  fall  of  potential,  or  difference  of  potential 
between  the  plates  of  the  cell. 

400.  Deflection  of  a  Magnet  by  a  Current  —  Hold  the  wire 
joining  the  plates  of  a  simple  voltaic  cell  over  and  parallel  to  a  pivoted 
magnetic  needle,  Fig.  316,  and 
then  close  the  circuit  by  placing 
the  plates  in  the  liquid,  or  by 
means  of  a  key  inserted  anywhere 
in  the  circuit.  The  needle  will  be 
turned  on  its  pivot  and  finally  come 
to  rest  at  an  angle  with  the  con>- 
ductor  carrying  the  current.  If  FIG.  316.- A  Magnetic  Needle  is  De- 

/,    *  fleeted  by  an  Electric  Current, 

the  current  be  passed  in  the  op- 
posite direction  above  the  needle,  the  deflection  is  opposite  to  the 
first.    The  wire  may  now  be  placed  below  the  needle,  and  the  direction 
of  the  deflections  obtained.     If  the  conductor  be  placed  at  the  side, 
or  near  the  end,  Of  a  suspended  magnet,  deflections  are  also  obtained. 

This  experiment  confirms  the  result  previously  found 
in  §  390,  viz.  that  the  region  around  a  wire  through  which 
an  electric  charge  is  moving  has  magnetic  properties. 
This  experiment  was  first  performed  by  Oersted,1  a  Danish 
physicist,  in  1819.  Oersted's  discovery  is  of  great  his- 
torical interest,  since  it  was  the  first  evidence  obtained  in 
regard  to  the  magnetic  effect  of  a  current  of  electricity 
and  has  led  to  results  of  the  greatest  practical  importance, 
1  See  portrait  facing  page  382. 


382 


A  HIGH   SCHOOL   COURSE   IN   PHYSICS 


FIG.  317.  —  The  Thumb  Shows  the 
Direction  in  Which  the  N-Pole 
Moves. 


By  observing  the  direction  in  which  the  N-pole  of  the 
magnetic  needle  is  moved  in  relation  to  the  direction  of 
the  electric  current,  the  following  rule  will  be  found  to 

apply  :  Let  the  fingers  of  the  out- 
stretched right  hand  point  in  the 
direction  of  the  current  flow  in 
the  wire  and  the  palm  be  turned 
toward  the  needle;  the  extended 
thumb  will  then  show  the  direction 
of  the  deflection  of  the  N-pole  of 
the  needle.  The  rule  is  of  convenience  in  determining  the 
direction  of  a  current  when  its  effect  upon  a  magnetic 
needle  is  known.  The  manner  of  applying  this  rule  is 
made  clear  by  Fig.  317. 

401.  The  Galvanometer.  —  The  effect  discovered  by 
Oersted  is  of  great  service  in  the  galvanometer,  an  instru- 
ment used  for  detecting  and  measuring  electric  currents. 
The  following  experiment  will  show  how  the  effect  of  a 
current  on  a  magnetic  needle  can  be  so  increased  as  to 
make  it  possible  to  discover  even  very  feeble  currents. 

Place  a  compass  needle,  below  which  is  a  graduated  circle,  within 
a  single  turn  of  wire  as  in  (1),  Fig.  318,  and  read  the  deflection  on  the 
scale.     Now  wind  the  same  wire  three  or  four  times  around  the  com- 
pass, always  keeping  the  wire  parallel  to 
the  original  position  of  the  needle,  and 
again  read  the   deflection.     The   second 
reading  will  be   much  greater  than  the 
first.     A  current  that  will  scarcely  move 
the  needle  when  a  single  turn  of  wire  is 
used  will  be  found  to  produce  a  marked 
effect  when  tested  with  the  coil  of  several 
turns. 

The  magnetic  forces  due  to  the 
electric  current  in  all  parts  of  the 
coil  tend  to  turn  the  needle  in  one  eter. 


HANS    CHRISTIAN    OERSTED    (1777-1851) 


Oersted's  famous  experiment  of  1819  on 
the  deflection  of  a  magnet  by  an  electric 
current  was  the  beginning  of  the  science  of 
electro-magnetism.  This  experiment  dem- 
onstrated the  long-sought  connection  be- 
tween electricity  and  magnetism  and  served 
to  point  out  the  line  of  experimentation 
that  has  brought  the  science  up  to  its  pres- 
ent-day development. 

Oersted  was  born  in  Langeland,  a  por- 
tion of  Denmark,  studied  at  Copenhagen, 
and  afterwards  became  a  professor  at  the 
university  and  polytechnic  schools  of  that 
city. 


DOMINIQUE    FRANCOIS    JEAN    ARAGO     (1786-1853) 


The  year  following  Oersted's  discovery, 
Arago,  a  noted  Parisian  astronomer  and 
physicist,  observed  that  iron  filings  cling  to 
a  conductor  carrying  an  electric  current. 
A  little  later  it  was  shown  by  Sir  Humphry 
Davy  of  England  that  the  filings  arrange 
themselves  in  magnetized  chains  around  the 
conductor. 

Arago  was  one  of  the  first  advocates  of 
the  wave  theory  of  light.  The  beautiful 
tints  produced  when  polarized  light  passes 
through  certain  crystals  were  discovered  by 
him  in  1811.  Arago  planned  a  method  for 
measuring  directly  the  velocity  of  light  in 
air  and  water,  but  failing  eyesight  pre- 
vented carrying  out  his  experiments.  He  lived,  however,  to  see  the 
work  done  by  Fizeau  and  Foucault. 


VOLTAIC   ELECTRICITY  383 

particular  direction,  a  fact  that  becomes  clear  when  the 
rule  given  in  §  400  is  applied.  Hence  by  introducing 
a  sufficient  number  of  turns  of  wire  and  making  the 
needle  extremely  light,  a  very  small  current  will  suffice 
to  produce  a  deflection. 

402.  Polarization  of  a  Voltaic  Cell.  —  i.  Connect  a  simple 
zinc  and  copper  cell  with  a  voltmeter1  (§  430)  or  a  high-resistance 
galvanometer  and  read  the  deflection  produced.  Short-circuit  the  cell 
for  a  short  time  by  means  of  a  wire,  thus  allowing  hydrogen  to  form 
in  large  quantities  at  the  copper  plate,  remove  the  wire,  and  again 
read  the  deflection.  It  will  be  less  than  at  first.  If  the  liquid  is 
now  stirred,  the  deflection  will  be  increased.  Keep  the  bubbles 
brushed  from  the  copper  plate  and  see  if  the  current  can  be  kept  the 
same  as  at  first. 

2.  Substitute  a  carbon  plate  for  the  copper  and  repeat  Experi- 
ment 1.  (A  good  carbon  plate  may  be  taken  from  a  discarded  dry  cell.) 
A  similar  decrease  in  the  current  will  be  observed.  While  the  cell  is 
connected  with  the  instrument,  pour  into  the  sulphuric  acid  a  small 
quantity  of  sodium  (or  potassium)  dichromate  or  chromic  acid  solu- 
tion. The  index  of  the  galvanometer  promptly  indicates  a  strong 
increase  of  electromotive  force.  The  cell  may  now  be  short-circuited 
as  before,  but  the  E.  M.  F.  quickly  resumes  its  original  value  after 
the  short  circuit  is  removed. 

These  experiments  show  clearly  that  the  collection  of 
hydrogen  on  the  copper  (or  carbon)  plate  of  a  cell  reduces 
the  E.  M.  F.  and,  consequently,  the  current  that  it  sends 
through  the  conductor.  This  effect  arises  from  the  fact 
that  a  hydrogen-coated  plate  now  takes  the  place  of  the 
copper  plate.  The  diminution  of  the  E.  M.  F.  of  a  cell 
by  the  presence  of  hydrogen  on  the  copper  (or  carbon)  plate 

1  For  classroom  demonstration  purposes  it  is  desirable  that  a  volt- 
meter be  provided  having  upwards  of  50  ohms  resistance  and  reading 
to  about  5  volts.  An  ammeter  having  less  than  an  ohm  resistance  and 
reading  to  3  or  4  amperes  will  also  be  found  of  great  service.  In  case 
such  instruments  are  not  available,  a  galvanometer  of  large  resistance  (50 
ohms  or  more)  may  be  substituted  for  a  voltmeter,  and  one  of  small 
resistance  (less  than  1  ohm),  for  an  ammeter. 


384 


A  HIGH   SCHOOL  COURSE  IN   PHYSICS 


is  called  the  polarization  of  the  cell.  If  the  bubbles  be 
removed  by  stirring,  the  current  will  remain  near  its 
original  value.  Experiment  2  serves  to  demonstrate  the 
fact  that  the  effect  of  the  hydrogen  can  be  largely  overcome 
by  chemical  means,  a  method  of  which  advantage  is  taken 
in  many  kinds  of  voltaic  cells.  In  the  prevention  of 
polarization  by  chemical  action  the  dichromate  acts  as  a 
depolarizer  for  removing  the  hydrogen  from  the  carbon 
plate.  This  it  does  by  supplying  an  abundance  of  oxygen, 
with  which  the  hydrogen  unites  chemically  to  form 
water. 

403.  The  Dichromate,  or  Grenet,  Cell.  —  In  this  cell  a 
zinc   plate  Z,   Fig.    319,   is  usually  placed  between  two 

plates  of  carbon  which  are  joined  together 
by  metal  at  the  top.  The  liquid  is  dilute 
sulphuric  acid  (H2SO4)  to  which  is  added 
dichromate  of  sodium  (or  potassium)  or 
chromic  acid  as  a  depolarizer.  See  §  402, 
Exp.  2. 

The  dichromate  cell  is  capable  of  giving 
a  strong  current  for  a  short  time  and  for 
this  reason  has  been  largely  used  in  ex- 
perimental work.  It  has,  however,  been 
largely  replaced  by  the  "  dry  "  cell  (§  407) 
and  storage  battery  on  account  of  their 
greater  convenience.  One  disadvantage  of 
the  dichromate  cell  is  the  necessity  of  with- 
drawing the  zinc  from  the  acid  by  the  rod 
A  when  the  cell  is  not  in  use. 

404.  The   Daniell   Cell. —The   Daniell  cell,   Fig.    320, 
consists  of  a  glass  jar  containing  a  saturated  solution  of 
copper  sulphate  (blue   vitriol,  CuSO4)   in  which    stands 
a  large  sheet  copper  plate  O.     The  copper  plate  encircles 
a  porous  cup  of  unglazed  earthenware  which  contains  a 


FIG.  319.  —  The 
Grenet,  or 
Dichromate 
Cell. 


VOLTAIC  ELECTRICITY 


FIG.   320.  —  The 
Daniell  Cell. 


heavy  bar  of  zinc  Z  immersed  in  a  dilute  solution  of  zinc 
sulphate  (ZnSO4).  The  porous  cup  does  not  check  the 
flow  of  electricity,  but  does  prevent  the  rapid  mixing  of 
the  two  solutions.  Dilute  sulphuric  acid  may  be  used 
in  place  of  zinc  sulphate. 

In  this  cell  the  zinc  is  continually  being  dissolved  and 
in  the  course  of  time  must  be  renewed.  On  the  other 
hand,  copper  (Cu)  from  the  solution  of 
copper  sulphate  (CuSO4)  is  deposited  slowly 
upon  the  copper  plate,  which  in  time  grows 
into  a  massive  sheet.  In  order  to  maintain 
c,  constant  supply  of  copper  ions  in  the 
solution,  crystals  of  copper  sulphate  are 
added  from  time  to  time.  Since  copper, 
instead  of  hydrogen,  is  deposited  on  the 
copper  plate,  the  nature  of  the  plate'  is 
not  changed,  and,  consequently,  no  polarization  takes 
place.  On  account  of  the  complete  non-polarization  of  the 
Daniell  cell,  it  is  often  used  when  currents  of  great  con- 
stancy are  required. 

405.  The  Gravity  Cell.  —  A  dilute  solution  of  zinc 
sulphate  has  less  density  than  a  saturated  solution  of 
copper  sulphate;  hence,  the  two  will  be 
kept  separate  by  gravity  and  the  former 
will  .float  upon  the  latter.  This  fact  is 
employed  in  the  so-called  gravity  cell,  Fig. 
321,  in  which  a  copper  plate  lies  upon  the 
bottom  of  a  jar  surrounded  by  crystals  of 
copper  sulphate  and  a  saturated  solution 
of  the  same  substance.  Above  this  solu- 
tion is  one  of  dilute  zinc  sulphate  which  surrounds  a 
massive  zinc  plate. 

If  a  gravity  cell  be  allowed  to  stand  on  an  open  circuit, 
the  liquids  slowly  mix.     In  order  to  prevent  this  and  thus 
26 


FIG.    321.  —  The 
Gravity  Cell. 


386         A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

keep  the  copper  sulphate  from  reaching  the  zinc  plate,  the 
cell  must  be  kept  in  more  or  less  active  operation.  While 
producing  a  current  the  copper  ions  are  continually  mov- 
ing away  from  the  zinc,  in  which  respect  the  action  is  the 
same  as  that  of  the  Daniell  cell,  of  which  this  cell  is 
a  modification.  Gravity  cells  are  extensively  used  in 
telegraphy  and  in  circuits  where  constant  currents  are 
desired. 

406.    The  Leclanche  Cell.  — The  positively  charged  plate 
of  the  Leclanche  (pronounced  Le  dan'shd1)  cell,  Fig.  322, 
is  a  bar  of  carbon  C  which  is  packed  in 
a  porous  cup  together  with  small  pieces 
of  carbon  and  manganese  dioxide.     The 
porous   cup   is   placed   in    a    solution   of 
-  ammonium     chloride     (salammoniac)     in 
which  stands  a  bar  of  zinc  Z  to  serve  as 
the  negative  plate  of  the  cell. 

When  the  circuit  containing  a  Leclanche 
FIG.  322.— The  Le-    CQ\\  js  closed,  hydrogen  is  liberated  at  the 

clanche'  Cell.  J 

carbon ;  but,  on  account  of  the  presence 
of  the  manganese  dioxide,  the  hydrogen  is  slowly  oxidized, 
forming  water.  In  this  manner  polarization  is  largely 
prevented.  As  a  rule,  however,  the  hydrogen  is  liberated 
so  rapidly  that  the  cell  slowly  polarizes,  but  regains  its 
normal  condition  when  allowed  to  stand  for  a  time  on 
an  open  circuit. 

The  Leclanche  cell  has  had  a  very  extensive  use  on 
account  of  the  fact  that  it  produces  currents  that  are 
suitable  for  ringing  bells,  operating  signals,  regulating 
dampers,  etc.  The  cell  will  remain  in  good  condition  for 
years  with  very  little  attention.  At  the  present  time  it 
is  being  rapidly  replaced  by  the  more  convenient  and  in- 
expensive "dry"  cell,  which  is  a  modified  form  of  the 
Leclanche. 


VOLTAIC  ELECTRICITY 


387 


407.  The  "Dry  "  Cell.  —  The  Leclanche  cell  is  made  in 
the  form  of  the  so-called  "  dry  "  cell  by  embedding  the  car- 
bon plate  (7,  Fig.  323,  in  a  paste  A  made  by  mixing  zinc 
oxide,  ammonium  chloride, 
plaster  of  Paris,  zinc  chlo- 
ride, and  water.  The  whole 
mass  is  contained  in  a  zinc 
cup  Z,  which  serves  as  the 
negative  plate  of  the  cell. 
Evaporation  is  prevented  by 
hermetically  sealing  the  cup 
with  melted  bitumen  or 
asphalt.  Many  different 
forms  of  dry  cells  are  now 
on  the  market  and  are  in 
great  demand  for  operating 
the  sparking  devices  of  gas 
and  gasoline  engines,  ring- 
ing bells,  etc.  A  "  dry "  cell  deteriorates  rapidly  on  a 
closed  circuit,  and  hence  should  always  be  connected  with 
a  spring  key  that  automatically  opens  the  circuit  when 
the  cell  is  not  in  use. 

EXERCISES 

1.  Explain  how  the  direction  of  the  current  in  a  telegraph  wire 
could  be  determined  by  means  of  a  small  compass. 

2.  Is  the  difference  of  potential  between  the  plates  of  a  voltaic  cell 
large  enough  to  cause  a  spark  when  wires  from  them  are  brought 
near  together  ?    Is  it  great  enough  to  produce  a  shock  when  the  plates 
are  simultaneously  touched? 

3.  Would  you  expect  to  get  an  E.  M.  F.  by  forming  a  cell  of  two 
copper  plates  or  two  zinc  plates  in  dilute  sulphuric  acid  ?    Make  the 
experiment,  using  a  sensitive  galvanometer.     Try  the  experiment  with 
a  polarized  copper  plate  and  one  that  is  not  polarized  and  account  for 
the  results. 

4.  Would  you  expect  to  derive  a  current  from  a  zinc  and  copper 
cell  containing  a  solution  of  common  salt?    Perform  the  experiment 


FIG.  323.  —  The  "Dry  "  Cell. 


388 


A  HIGH   SCHOOL  COURSE  IN   PHYSICS 


5.  Why  is  the  combination  of  zinc,  copper,  and  dilute  sulphuric 
acid  a  suitable  one  for  experiments  like  those  described? 

6.  Study  the  description  of  the  Daniell  cell  and  state  which  solu- 
tion grows  weaker  and  must  ultimately  be  renewed.     What  materials, 
therefore,  must  be  kept  on  hand  for  replenishing  a  system  of  Daniell 
or  of  gravity  cells  ? 

2.     EFFECTS  OF  ELECTRIC  CURRENTS 

408.  The  Magnetic  Effect.  —It  has  already  been  shown 
that  a  current  of  electricity  is  capable  of  magnetizing  bars 
of  iron  and  steel  (§  390),  and  also  that  a  magnetic  needle 
placed  near  a  current  is  deflected  from  its  normal  position. 
These  are  only  special  cases  of  the  more  general  one  illus- 
trated by  the  following  experiments: 

1.  Join  several  new  dry  cells  in  series  and  connect  them  through 
a  key  to  a  vertical  wire  passing  through  a  horizontal  sheet  of  card- 


FIG.  324. —  Magnetic  Field 
around  a  Conductor  Carrying 
a  Current. 


FIG.  325.  —  Iron  Filings  Arranged 
around  a  Conductor. 


board  as  in  Fig.  324.  A  current  of  at  least  2  amperes  is  desirable. 
Sprinkle  iron  filings  on  the  cardboard  and  close  the  key  for  a  short 
time,  meanwhile  tapping  lightly  to  jar  the  filings.  The  filings  will 
arrange  themselves  in  circular  lines  (see  Fig.  325)  around  the  wire 
as  the  center,  thus  showing  the  shape  of  the  magnetic  field  about  the 
conductor.  Small  pivoted  magnets  placed  near  the  wire  will  turn 


VOLTAIC  ELECTRICITY  389 

until  they  are  tangent  to  the  circles  with  their  N-poles  as  shown  in 
Fig.  324. 

2.  Make  a  helix,  or  spiral,  of  insulated  copper  wire  by  winding 
about  30  turns  upon  a  lead  pencil.     Connect  the  ends  of  the  wire  with 
a  cell  and  present  one  end  of  the  helix  to  the  N-pole  of  a  suspended 
magnet  or  compass  needle,     One  end  will  be  found  to  attract  the 
N-pole  while  the  opposite  end  repels  it,  and  the  end  that  attracts 
the  N-pole  will  repel  the  S-pole.     Reverse  the  connections  of  the  cell 
and  repeat  the  experiment.     The  end  of  the  helix  which  formerly 
attracted  a  magnetic  pole  will 

now  repel  it. 

3.  Thread  a  spiral  of  cop- 
per wire  through  holes  in  a 
flat  piece  of  cardboard  or  wood 
as  shown  in  Fig.  326.     Strew 

iron    filings    evenly    over    the    FIG.  326.  — Magnetic  Field  Produced  by  a 

f  ,    ,1  Current  in  a  Helix,  or  Solenoid, 

surface  and  send   the  current 

(about  3  amperes  are  required)  from  several  new  dry  cells  through 
the  wire.  If  the  apparatus  be  tapped  lightly,  the  filings  will  arrange 
themselves  as  shown  in  the  figure. 

It  is  clear  from  these  experiments  that  a  conductor 
carrying  an  electric  current  is  surrounded  by  a  magnetic 
field.  If  the  conductor  is  a  straight  wire,  Experiment  1 
shows  that  the  lines  of  force  are  concentric  circles  around 
the  conductor  as  the  center.  The  direction  of  the  lines  is 
given  by  the  direction  in  which  an  N-pole  is  urged  when 
placed  in  the  field.  It  is  clear,  therefore,  from  Fig.  324, 
that  the  direction  of  the  lines  is  that  indicated  by  the 
arrows.  A  convenient  rule  may  be  stated  as  follows : 

G-rasp  the  conductor  with  the  right  hand  with  the  out- 
stretched thumb  in  the  direction  the  current  is  flowing.  The 
fingers  encircle  the  wire  in  the  direction  of  the  lines  of  force. 

The  shape  of  the  field  depends  on  the  form  of  the  con- 
ductor ;  for,  when  the  conductor  is  a  helix,  the  magnetic 
field  resembles  that  about  a  straight  bar  magnet.  (See 
Fig.  327.)  In  fact,  it  is  shown  in  Experiment  2  that  the 


390 


A  HIGH   SCHOOL   COURSE   IN   PHYSICS 


helix  has  the  properties  of  a  magnet  while  a  current  is 
flowing  through  the  wire.  If  the  coil  could  be  properly 
suspended  and  a  current  sent  through  it,  the  axis  would 
assume  the  direction  taken  by  a  compass  needle.  Such  a 
helix  is  also  called  a  solenoid. 


FIG.  327.  —  Showing  the  Relation  between  the  Direction  of  the  Current  and 
the  Magnetic  Lines. 

409.  Poles  of  a  Helix.  —  Let  Experiment  2  of  §  408  be  repeated 
and  the  N-pole  and  the  S-pole  ascertained.  Tracing  the  current  from 
the  positive  pole  of  the  cell,  it  will  be  found  to  flow  around  the  N-pole 
in  a  direction  contrary  to  the  motion  of  the  hands  of  a  clock  as  one 
faces  the  pole  and  in  the  reverse  direction  about  the  S-pole  as  shown 
in  Fig.  327. 

The  experimental  result  leads  to  the  following  conven- 
ient rule :  If  the  helix  be  grasped  with  the  right  hand  so  that 

the  fingers  point  in  the  direction 
the  current  is  flowing,  the  ex- 
tended thumb  will  point  in  the 
direction  of  the  N-pole  of  the 
helix.  (See  Fig.  328.) 

Another  convenient  rule  is 
applied  by  facing  the  end  of  the 
helix ;  if  the  current  is  flow- 
ing clockwise,  the  end  of  the  helix  is  an  S-pole,  and, 
conversely,  if  counter-clockwise,  an  N-pole. 


FIG.  328.  — Rule  for  Determining 
the  Poles  of  a  Helix  Carrying 
a  Current. 


VOLTAIC  ELECTRICITY  391 

410.  The  Electro-magnet.  —  Probably  none  of  the  effects 
that  can  be  produced  by  electric  currents  are  of  greater 
practical  value  or  employed  more  extensively  than  the 
magnetic  effect.  The  scale  on  which  this  effect  can  be 
produced  is  limited  only  by  the  dimensions  of  the  appa- 
ratus used  and  the  strength  of  the  current. 

Wind  a  helix  consisting  of  about  75  turns  of  No.  22  insulated 
copper  wire  and  provide  an  iron  core  that  can  be  inserted  into,  or 
removed  from,  the  helix  as  de- 
sired. Test  the  magnetic  action 
of  the  helix  without  the  core 
by  presenting  it  to  a  magnetic 
needle  while  a  current  is  flow- 
ing. Now  insert  the  core  and 
note  the  change.  Its  effect  is 
more  marked  than  before.  Next 
send  the  current  from  a  new  Fia'  329.  -  Magnetization  of  Iron  by 
,  „  ,.  ,  ,.  ,  ,.  .  ,  Means  of  an  Electric  Current, 

dry  cell  through  the  helix  and 

dip  one  end  of  the  core  into  a  box  of  tacks,  Fig.  329.  A  large 
quantity  of  tacks  will  cling  to  the  core  and  remain  there  until  the 
current  is  interrupted. 

Although  a  coreless  helix  of  wire  has  magnetic  proper- 
ties when  a  current  is  flowing  through  it,  its  magnetic 
field  is  insignificant  when  compared  to  that  which  the 
same  current  will  produce  when  an  iron  core  is  present, 
on  account  of  the  large  permeability  of  iron  (§  381). 
The  introduction  of  the  core  adds  the  lines  of  force  of 
the  magnetized  iron  to  those  produced  by  the  current  in 
the  helix  alone.  Any  mass  of  iron  around  which  is  a  helix 
for  conducting  an  electric  current  is  called  an  electro-magnet. 
The  small  amount  of  magnetism  retained  by  the  core  after 
the  circuit  is  broken  is  termed  residual  magnetism. 

Electro-magnets  are  made  in  many  forms  and  often  of 
extremely  large  dimensions  for  holding  heavy  masses  of 
iron.  The  horseshoe  form  shown  in  Figs.  330  and  331  is 


392 


A  HIGH   SCHOOL   COURSE   IN   PHYSICS 


most  frequently  used  in  electrical  devices.  The  wire  is 
so  wound  as  to  produce  an  N-pole  at  N  and  an  S-pole 
at  S.  The  bar  of  iron  A  which  is  held  by  the  magnetism 
of  the  poles  is  called  the  armature.  When 
the  armature  is  against  the  poles,  it  will 
be  observed  that  the  lines  of  force  find  a 


i 


FIG.  330.— An  Electro-magnet 
Showing  Poles  and  Armature. 


FIG.  331.  —  Showing 
the  Path  of  the 
Lines  of  Force  in 
an  Electro-magnet. 


FIG.  332.  —  A  Large 
Electro-magnet 
Used  for  Han- 
dling Masses  of 
Iron  in  Factories. 


complete  magnetic  circuit  through  iron,  as  shown  by 
the  dotted  lines  in  Fig.  331.  Figure  332  shows  a  large 
form  of  the  electro- magnet  that  is  widely  used  in  manu- 
facturing plants  for  the  purpose  of  handling  heavy  masses 

of  iron,  as  castings, 
plates,  pig  iron,  etc. 
The  lifting  power  is  con- 
trolled mainly  by  the 
current  used. 

411.   The  Electric  Bell. 

—  An  important  application 
of  the  magnetic  effect  of  an 
electric  current  is  found  in 
the  electric  bell.  The  instru- 
ment consists  of  an  electro- 
magnet %,  Fig.  333,  near  the 
poles  of  which  is  an  armature 
of  iron  attached  to  a  spring. 
Extending  from  the  armature 
FIG.  333.  —  The  Electric  Bell.  is  a  slender  rod  bearing  at  its 


VOLTAIC  ELECTRICITY 


393 


extremity  the  bell  hammer  H.  The  armature  carries  a  spring  that 
touches  lightly  against  the  screw  point  at>C.  The  connections  are  made 
as  shown  in  the  figure.  When  the  push  button  P  is  pressed,  the  circuit 
is  completed  by  a  metallic  contact  within  the  button,  and  the  current 
from  the  cell  B  flows  through  the  electro-magnet  coils.  This  causes 
the  magnet  to  attract  the  armature,  and  the  hammer  strikes  the  bell. 
The  movement  of  the  armature,  however,  breaks  the  circuit  at  C  and 
thus  interrupts  the  current.  Since  the  cores  of  the  magnet  now  lose 
their  magnetism,  the  armature  is  thrown  back  by  the  spring ;  the 
contact  at  C  is  restored,  and  all  the  operations  are  repeated.  Hence 
a  steady  pressure  on  the  push  button  causes  the  hammer  to  execute  a 
number  of  rapid  strokes  against  the  bell.  The  direction  of  the  cur- 
rent is  immaterial  to  the  operation  of  the  bell. 

412.  Mutual  Action  of  Two  Parallel  Currents.  —  Suspend 
two  light  wires  about  30  cm.  long  from  two  other  wires  a  and  b, 
Fig.  334,  which  are  bent  as 
shown.  Let  the  lower  ends  of 
the  vertical  wires  just  dip  into 
a  mercury  cup  below.  Now  send 
a  strong  current  through  the  ap- 


FIG.  334.  —  Illustrating  the  Mutual 
Action  between  Parallel  Currents. 


FIG.  335.  —  Parallel  Currents  in  the 
Same  Direction. 


paratus  (an  alternating  current  of  from  4  to  8  amperes  suffices),  which 
will  flow  down  one  wire  and  up  the  other.  The  two  conductors  will 
repel  each  other.  Again,  hang  both  wires  on  a  and  make  the  con- 
nections as  shown  in  Fig.  335.  Both  currents  will  now  flow  in  the 
same  direction  and  an  attraction  will  be  observed. 


394 


A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


The  results  of  the  experiments  may  be  stated  as  follows: 
Currents  flowing  in  the  same  direction  attract  each  other, 
and  those  flowing  in  opposite  directions  repel  each  other.  The 


FIG.  336.  — Magnetic  Fields  around  Parallel  Conductors :  (1)  When  Currents 
are  Flowing  in  the  Opposite  Directions ;  (2)  When  Flowing  in  the  Same 
Direction. 

magnetic  field  around  the  two  wires  in  each  of  the  two 
cases  is  shown  in  Fig.  336. 

413.  Heating  Effect  of  Electric  Currents.  —  It  is  a  familiar 
fact  that  electric  currents  produce  heat.  This  is  at  once 
evident  when  the  hand  is  placed  in  contact  with  the  warm 
bulb  of  an  incandescent  lamp.  The  following  experiment 
shows  in  another  way  the  transformation  of  electrical 
energy  into  heat. 

If  an  electric  lighting  current  is  available,  construct  a  device  for 
controlling  it  by  mounting  several  lamp  sockets  on  a  board  and  join- 
ing them  in  parallel  (§444).  Connect  each  side  to  a  binding  post, 
screw  a  lamp  into  each  socket,  and  a  safe  adjustable  resistance  is 
ready  for  use.  Put  the  device  just  described  in  circuit  with  a  piece 
of  fine  iron  wire  joined  to  one  of  copper  wire  of  the  same  diameter. 
The  pieces  may  be  each  several  inches  in  length.  Screw  one  lamp  at 
a  time  into  its  socket  until  sufficient  current  flows  to  heat  the  wire. 
The  iron  wire  will  at  length  glow  with  heat  while  the  copper  wire  of 
the  same  dimension  is  still  comparatively  cool.  If  a  lighting  current 
cannot  be  used,  the  experiment  may  be  made  by  using  shorter  pieces 
of  wire  with  several  good  dry  cells  joined  in  series.  In  this  case  no 
controlling  resistance  is  necessary. 

An  experiment  that  succeeds  well  with  a  small  current  is  made  by 
connecting  a  cell  with  a  fine  insulated  copper  or  iron  wire  that  has 


VOLTAIC  ELECTRICITY 


395 


been  wound  in  a  coil  about  the  bulb  of  a  thermometer, 
mercury  indicates  the  heating  effect  of  the  current. 


The  rise  of 


The  heating  effect  of  a  current  of  electricity  is  employed 
in  electric  cooking  devices,  in  various  methods  of  heating, 
and  in  electric  lighting. 
An  electrically  heated 
flatiron  is  shown  in  Fig. 
337. 

414.  Chemical  Effects 
of    Electric  Currents. — 

Seal  two  platinum  wires  to 

which  are  attached  platinum  FIG.  337.— An  Electrically  Heated  Flatiron. 
strips  about  1  cm.  wide  and 

3  cm.  long  in  small  bent  glass  tubes  e  and  /,  Fig.  338.     Over  each 
of  these  place  an  inverted  test  tube  filled  with  water  to  which  has 
been  added  a  number  of  drops  of  sulphuric   acid.     Place   a  small 
.   quantity  of  mercury  in  each  of  the  small 
tubes  and  connect  the  strips  of  platinum 
in  circuit  with  three  or  four  dry  cells 
joined   in  series  by  inserting  the  con- 
necting wires  into  the  mercury.     Small 
bubbles  of  gas  will  be  seen  rising  from 
the  platinum  in  each  tube  and  collect- 
ing at  the  top.     If  e  is  joined  to  the 

FIG.    338.  — Electrolysis   of    positive   pole   of   the   battery,  the   gas 
al tand%y!^nCa°"red    »bove  that  strip  of  platinum  will  collect 
only  one  half  as  fast  as  that  over  the 

other.  When  the  action  has  progressed  until  one  of  the  tubes 
is  filled  with  gas,  remove  it  carefully  and  apply  a  lighted  match. 
A  blue  flame  will  be  seen  caused  by  the  burning  of  this  gas,  which  is 
hydrogen.  Now  remove  the  other  tube  and  insert  a  glowing  pine 
stick,  which  will  be  observed  to  burst  into  a  flame  because  of  the 
oxygen  contained  in  the  tube. 

The  platinum  strips  employed  in  making  electrical  con- 
tact with  the  liquid  in  the  tubes  are  called  electrodes. 
While  conducting  the  electric  current  from  one  electrode 
to  the  other,  the  water  is  decomposed  into  its  constituent 


396         A  HIGH  SCHOOL  COURSE  IN  PHYSICS 

elements,  hydrogen  and  oxygen.  This  decomposition  is 
characteristic  of  all  liquids,  except  liquid  metals.  The 
process  of  decomposing  a  compound  substance  by  means  of 
an  electric  current  is  called  electrolysis.  The  substance 
decomposed  is  called  an  electrolyte.  The  electrode  at 
which  the  current  enters  the  electrolyte  is  the  anode; 
the  one  at  which  it  leaves,  the  cathode. 

415-  Theory  of  Electrolysis.  —  In  the  light  of  the  theory  of 
solutions  stated  in  §  396,  the  process  of  electrolysis  is  easily  explained. 
When  a  small  quantity  of  sulphuric  acid  (H2SO4),  for  example,  is 
introduced  into  water,  the  molecules  split  up  into  hydrogen  (H)  ions 
and  sulphions  (SO4),  the  former  bearing  positive  charges  of  electric- 
ity, the  latter,  negative  charges.  Now  when  two  platinum  electrodes 
that  are  connected  to  the  poles  of  a  battery  are  placed  in  the  liquid, 
the  cathode,  which  is  charged  negatively,  attracts  the  positively 
charged  H  ions,  while  the  anode,  which  is  positively  charged,  attracts 
the  negatively  charged  SO4  ions.  The  H  ions  discharge  their  positive 
electricity  to  the  cathode  and  are  then  free  to  collect  and  rise  in 
bubbles  to  the  surface.  The  SO4  ions,  on  the  other  hand,  discharge 
their  negative  electricity  to  the  anode,  where  they  react  chemically 
upon  the  water  (H2O),  setting  free  the  oxygen  (O).  Thus  each 
sulphion  (SO4)  together  with  the  hydrogen  (H2)  taken  from  the 
water  forms  new  molecules  of  sulphuric  acid  (H2SO4) .  In  this  manner 
the  amount  of  acid  present  in  the  solution  remains  constant,  while  the 
quantity  of  water  diminishes  as  the  process  of  electrolysis  advances. 

416.  Electrolysis  of  Copper  Sulphate.  —  Introduce  two  plati- 
num electrodes  into  a  solution  of  copper  sulphate  (CuSO4)  and  place 
them  in  circuit  with  three  or  four  dry  cells.  After  a  few  seconds  the 
cathode,  or  negative  electrode,  will  be  found  to  be  coated  with  me- 
tallic copper,  while  the  anode  remains  unchanged.  If  the  direction  of 
the  current  be  now  reversed,  copper  will  be  deposited  on  the  clean 
plate  (now  the  cathode),  while  the  copper  coating  on  the  anode  gradu- 
ally disappears. 

In  the  electrolysis  of  copper  sulphate,  the  ions  present  are  copper 
(Cu)  ions  and  sulphions  (SO4),  the  former  bearing  positive  charges 
of  electricity  and  the  latter,  negative.  When  platinum  electrodes  are 
introduced  into  the  solution,  the  copper  ions  are  attracted  to  the 
cathode,  where  they  discharge  their  electricity  and  become  free.  Thus 


VOLTAIC   ELECTRICITY 


397 


we  find  copper  deposited  upon  this  electrode.  At  the  anode  the  sul- 
phions  react  with  water  as  in  the  case  just  described  (§  415).  How- 
ever, if  the  anode  is  a  copper  plate,  the  sulphions  (SO4)  abstract  copper 
from  the  plate  at  the  instant  they  discharge  their  electricity  and  form 
copper  sulphate  (CuSO4),  which  dissolves  in  the  liquid. 

The  experiment  illustrates  the  process  of  electroplating, 
a  method  by  which  one  metal  is  given  a  coating  of  another. 
Thus  by  electrolytic  action  corrosive  iron  may  be  plated 
with  non-corrosive  nickel,  or  tarnishing  brass  with  pure 
gold. 

In  the  experiment,  copper  (Cu),  which  is  a  constituent 
of  the  copper  sulphate  (CuSO4)  in  the  solution,  is  always 
deposited  on  the  cathode,  or  negative  electrode.  If  the 
anode  is  made  of  platinum,  oxygen  will  be  liberated  at  its 
surface.  If,  however,  the  anode  is  made  of  copper,  the 
case  is  modified  ;  for,  instead  of  setting  oxygen  free, 
copper  is  removed  from  the  plate  and  carried  into  the 
solution. 

417.    Electroplating.  —  When  we  consider  the  vast  num- 
ber of  plated  articles  in  everyday  use,  we  can  scarcely 
overestimate  the  great  commercial  value  of  the  electrolytic 
action  of  an  electric  cur- 
rent.     When,    for   ex- 
ample,   silver  is  to  be 
plated  upon  the  surface 
of   a   spoon,  an   anode 
plate   of   silver  is  sus- 
pended in  a  solution  of 

.,  .  n  T  .,       FIG.  339.  —  Illustrating  the  Process  of  Elec- 

silver    cyanide,     while  tropiating. 

the  spoon  is  made  the 

cathode   by  being  connected  with  the  zinc  of  a  battery 

and   completely   submerged   in   the  solution.      See  Fig. 

339.     The  current  is  allowed  to  flow  until   the   coating 

is  of   the  desired   thickness.     In  the  nickel-plating  pro- 


398         A  HIGH   SCHOOL   COURSE   IN   PHYSICS 

cess  an  anode  of  nickel  is  used  in  a  solution  of  nickel 
nitrate  and  ammonium  nitrate.  The  article  to  be  plated 
is  always  the  cathode.  When  the  coating  reaches  the 
proper  thickness,  the  final  process  of  polishing  gives  the 
surface  the  desired  appearance. 

418.  Electrotyping.  — As  a  rule,  books  of  which  a  large 
edition    is  to  be  printed  are   first  electrotyped.     In  this 
process  an  impression  is  made  in  wax  after  the  type  has 
been  set  up,  so  that  each  letter  leaves  its  imprint  in  the 
mold.     A    thin  layer   of   finely  powdered  plumbago,  or 
graphite,  is  brushed  over  the  surface  of  the  wax  in  order 
to  render  it  a  conductor  of  electricity.     When  thus  pre- 
pared, the  mold  is  placed  in  an  electrolytic  bath  of  copper 
sulphate  and  joined  to  the  negative  pole  of  a  battery  or 
other  source  of  electricity.     The  anode  is  simply  a  copper 
plate.     The  current  is  allowed  to  flow  until  the  coating 
of  copper  upon  the  wax  is  somewhat  thicker  than  a  sheet 
of  paper.     While  the  copper  is  being  deposited  upon  the 
conducting  graphite,  it  penetrates  into  even  the  smallest 
depressions  of  the  mold  and  thus  reproduces  in  copper 
the  exact  form  of  the  type.     The  coating  of  copper  is 
then  removed  from  the  wax,  trimmed,  and  filled  in   at 
the  back  with  molten  type  metal.     The  advantage  gained 
by  electrotyping  is  convenience,  durability,  and  perma- 
nence, and  the  type  from  which  the  impression  is  taken 
on  the  wax  may  be  distributed  and  used  again  without 
delay. 

419.  Refining  Copper.  —  Copper  as  it  comes  from  ordi- 
nary smelting  works   contains    many  impurities.      Such 
copper  is   refined   electrolytically  by  casting   the   crude 
metal  in  huge  plates  which  are  afterwards  used  as  anodes 
in  large  depositing  vats.     The   solution   used  is  copper 
sulphate,  and  the  cathode  is  a  thin  plate  of  pure  copper. 
When  a  current  of  electricity  is  sent  through  the  solution, 


VOLTAIC  ELECTRICITY 


399 


copper  is  deposited  on  the  cathode  until  it  grows  into  a 
heavy  plate.  The  copper  anode  is  carried  into  the  solu- 
tion, while  its  impurities  collect  at  the  bottom  of  the  vat. 
Copper  thus  refined  is  called  electrolytic  copper,  and  is 
much  used  in  the  manufacture  of  wire  and  in  the  construc- 
tion of  dynamos,  motors,  etc. 

420.  The  Storage  Battery.  —  The  principle  of  the  storage  cell 
may  be  illustrated  by  making  a  small  cell  of  two  plates  of  lead  about 
2x6  inches  and  a  solution  of  sulphuric  acid  consisting  of  one  part  of 
acid  and  about  eight  parts  of  water.  Attach  the  plates  to  a  piece  of 
wood  and  hang  them  in  the  solution.  Connect  the  lead  plates  to  two 
good  dry  cells  joined  in  series  and  allow  the  current  to  flow  for  a  minute 
or  more.  Disconnect  the  dry  cells  and  run  wires  from  the  lead  plates 
to  an  electric  bell.  The  bell  will  ring  vigorously  for  a  short  time  and 
then  gradually  cease.  The  power  of  the  cell  can  be  restored  by  again 
connecting  it  with  the  dry  cells.  If  a  galvanometer  be  introduced  in 
the  circuit,  it  will  be  found  that  the  discharging  current  flows  in 
opposition  to  the  current  used  in  charging.  If  the  E.  M.  F.  of  the 
charged  plates  is  measured  by  a  voltmeter,  it  will  be  found  to  be 
about  two  volts. 

The  charging  current  decomposes  the  water  as  in  §  414.  Hydrogen 
is  liberated  at  the  cathode  plate;  but  the  oxygen  produced  at  the 
anode  changes  the  surface  of  the  plate  from  lead  (Pb)  into  lead  peroxide 
(PbO%),  which  may  be  recognized  by  its  brownish  color.  When  the 
plates  are  connected  with  the  electric 
bell,  a  current  flows  from  the  peroxide 
plate  through  the  bell  to  the  other, 
which  is  simply  lead.  The  current 
continues  until  the  thin  coating  is 
used  up. 

It  should  be  observed  that  the 
storage  cell  stores  energy  but  not  elec- 
tricity. The  work  done  by  the  charg- 
ing current  results  in  the  production 
of  the  energy  of  chemical  separation 
in  the  cell.  When  the  circuit  is 
closed,  this  amount  of  potonuiiul  energy 
is  transferred  to  the  bell,  where  it  ap- 
pears as  mechanical  energy  which  is 


340.  — A    Storage    Cell,  or 
Lead  Accumulator. 


400         A  HIGH   SCHOOL   COURSE   IN   PHYSICS 

finally  dissipated   as   sound  and  heat.     A  complete  storage  cell  is 
shown  in  Fig.  340. 

EXERCISES 

1.  What  would  be  the  effect  produced  upon  the  strength  of  the 
poles  of  a  bar  magnet  if  it  were  placed  in  a  helix  in  which  the  direc- 
tion of  the  current  around  the  N-pole  of  the  magnet  was  counter- 
clockwise ?    In  which  it  was  clockwise  ? 

2.  A  helix  is  suspended  so  as  to  turn  freely  in  a  horizontal  plane 
and  is  placed  above  a  strong  bar  magnet.     If  a  current  be  sent  through 
the  helix,  what  will  be  its  direction  around  that  end  which  stops  over 
the  S-pole  of  the  magnet  ? 

3.  Place  a  compass  box  upon  one  of  the  rails  of  an  electric  railway 
running  north  and  south  and  see  if  you  can  detect  the  presence  and 
direction  of  a  current. 

4.  Would  you  expect  a  compass  needle  to  point  north  and  south 
in  a  moving  trolley  car  ?    Why  ? 

5.  Which  is  most  readily  magnetized  when  placed  in  a  helix,  iron 
or   steel?     Which   will  retain    the   greater  amount  of  magnetism? 
How  could  you  produce  a  permanent,  magnet  by  the  help  of  a  dry 
cell  and  a  helix  ? 

6.  How  could  the  direction  of  an  electric  current  be  determined  by 
means  of  an  electrolytic  cell  through  which  it  could  be  caused  to  flow  ? 

7.  What  change  would  finally  occur  in  the  copper  sulphate  solu- 
tion in  an  electrolytic  cell  having  platinum  electrodes  if  the  current 
were  allowed  to  flow? 

8.  Would  the  result  in  Exer.  7  be  at  all  modified  if  the  anode 
were  copper  ?    Explain. 

9.  Electric  circuits  in  buildings  are  protected  against  too  strong 
currents  by  lead  wire  fuses  of  the  proper  size  placed  in  the  circuit. 
If  by  accident  the  current  becomes  strong  enough  to  be  unsafe,  the 
wire  melts.     Explain  in  full.     If  possible,  inspect  the  wiring  of  some 
building  and  report  on  the  form  in  which  the  fuses  are  made. 

SUMMARY 

1.  An   electric   current  is  a   continuous   discharge,    or 
movement,  of  electricity  (§  390). 

2.  Electric    currents    may  be   produced   by    chemical 
action,  as  in  voltaic  cells.     It  may  be  shown  that  one  of 
the  terminals  of  such  a  cell  is  charged  with  positive  elec- 
tricity, the  other  with  negative  (§§  391  and  392). 


VOLTAIC  ELECTRICITY  401 

3.  The  E.  M.  F.  of  a  cell  is  the  difference  of  potential 
between  these  charges  when  no  current  is  flowing  (§  393). 

4.  Much  of  the  potential  energy  of  the  zinc  of  a  cell 
is  wasted  by  "local  action."     This  may  be  largely  pre- 
vented by  amalgamation  (§  397). 

5.  A  current  flowing  in  a  conductor  near  and  parallel 
to  a  magnetic  needle  tends  to  deflect  it  from  its  position. 
This  principle  is  used  in  many  electrical  measuring  instru- 
ments, as  galvanometers,  etc.  (§  400). 

6.  The  E.  M.  F.  is  diminished  by  the  accumulation  of 
hydrogen  on  the  copper  (or  carbon)  plate.     This  effect 
is  known  as  polarization.   (§  402). 

7.  An  electric  current  is  surrounded  by  a  magnetic 
field.     When  the  conductor  carrying  the  current  encircles 
a  bar  of  iron  or  steel,  the  bar  becomes  a  magnet  (§  408). 

8.  The  electro-magnet  consists  of  a  helix  of  insulated 
wire  wound  upon  a  core  of  iron.     The  principle  of  the 
electro-magnet  is  employed  in  the  electric  bell  and  many 
other  important  devices  (§§  410  and  411). 

9.  Parallel    currents   flowing   in    the   same    direction 
attract  each  other,  and  those  flowing  in  opposite  directions 
repel  (§  412). 

10.  When   an   electric    current   flows   through  a  con- 
ductor, the  conductor  is  heated.     This  effect  is  applied  in 
electric   lighting,    in   heating   and    cooking  devices,  etc. 
(§  413). 

11.  An  electric  current  decomposes  water  into  hydro- 
gen and  oxygen,  hydrogen  being  liberated  at  the  cathode. 
When  a  current  flows  through  a  solution  of  a  metallic 
salt,  the  compound  is  decomposed  and  the  metal  liberated 
at  (or  plated  upon)  the  cathode.     This  effect  of  an  electric 
current  is  known  as  electrolysis  and  is  used  in  electro- 
plating, etc.  (§§  414  to  420). 

27 


CHAPTER   XIX 

ELECTRICAL  MEASUREMENTS 

1.    ELECTRICAL   QUANTITIES   AND    UNITS 

421.  Fundamental   Electrical   Magnitudes.  —  In    every 
electric   circuit   there   are   three  fundamental  quantities 
which  admit  of  measurement;  viz.   current  strength,  elec- 
trical resistance,  and  difference  of  potential.     The  first  of 
these,  current  strength,  may  be  likened  to  the  rate  at 
which  water  is  delivered  through  a  pipe  ;  electrical  resist- 
ance, to   the  friction  encountered   by  a  liquid  current ; 
and  difference  of  potential,  to  a  difference  of  level  (or 
pressure)  for  any  two  chosen  points  between  which  the 
current  is  flowing. 

422.  Current  Strength  —  the  Ampere. — In  many  of  the 
preceding  experiments  it  has  been  obvious  that  the  mag- 
nitude of  many  of  the  effects  produced  depended  on  a 
quantity   that   has   been   frequently   referred   to   as   the 
"strength  of  the  current."     For  some  effects  the  current 
must  be  strong,  for  others,  weak.     The  expression  refers 
to  the  rate  at  which  positive  electricity  is  being  discharged 
from  the  positive  to  the  negative  pole  through  the  circuit. 

The  unit  of  current  strength  is  the  ampere,  so  called  in 
honor  of  Ampere,1  a  French  physicist.  The  ampere  is  that 
current  which  will  deposit  in  an  electrolytic  cell  0.001118 
grams  of  silver  or  0.0003287  grams  of  copper  per  second. 
Of  the  currents  used  in  the  experiments  described  in  the 
preceding  sections,  the  largest  were  required  in  §  413  and 

1  See  portrait  facing  page  406. 
402 


ELECTRICAL   MEASUREMENTS 


403 


amounted  to  6  or  7  amperes.  As  a  rule,  the  current  used 
in  most  classroom  demonstrations  is  less  than  1  ampere  in 
value.  In  the  comparison  and  measurement  of  electric 
currents,  different  kinds  of  galvanometers  are  used. 

423.  The  Tangent  and  Astatic  Galvanometers.  —  The  tan- 
gent galvanometer  is  a  common  form  found  in   most  laboratories. 
It  consists  of  a  circular  coil  of  wire  wound  on  a  frame  about  30  cm. 
in    diameter.      See 

(1),  Fig.  341.     The 

ends    of    the     coil 

lead  to  binding 

posts   on  the  base. 

At   the    center    of 

the  coil  is   placed 

a    compass    needle 

below  which   is    a 

graduated      circle. 

When    in   use  the 

coil  of  the  instru- 
ment   is    placed 

north  and  south  and 

the  current  sent  through  the  coil.     The  instrument  derives  its  name 

from  the  fact  that  the  current  is  proportional  to  the  tangent  of  the  angle 

of  deflection.     If  the   current  necessary  to  deflect  the  needle  45°  is 

known,  other  currents  are  easily  measured. 

For  the  measurement  and  detection  of  small  currents  the  astatic  gal- 
vanometer (2),  Fig.  341,  is  sometimes  used. 
Two  similar  magnets  are  attached  to  a  ver- 
tical rod  so  that  their  N-poles  point  in  op- 
posite directions.  This  system  is  then 
suspended  so  that  the  lower  needle  swings 
within  a  coil  of  wire,  as  shown  in  Fig.  342. 
When  a  current  is  sent  through  the  coil,  all 
parts  of  it  tend  to  throw  the  needles  out  of 

FIG.  342.  —  The  Arrange-  the  north-and-south  positions.  This  form  of 
ment  of  the  Magnets  in  galvanometer  may  be  made  extremely  sen- 
an  Astatic  Galvanometer.  sitiye  to  gmaU  currents> 

424.  The  d'Arsonval  Galvanometer.  —  Galvanometers  of  the 
d'Arsonval   type   differ  from   those  described  in  §  423  in    that  the 


FIG.  341.  —  (7)  The  Tangent  Galvanometer; 
(2)   The  Astatic  Galvanometer. 


404 


A   HIGH   SCHOOL   COURSE   IN   PHYSICS 


magnet  is  stationary  and  the  coil  of  wire  movable.     As  shown   in 
Fig.  343,  the  instrument  consists  of  a  light  coil  of  many  turns  of  fine 

wire,  one  end  of  which  is  the 
supension  wire  AC,  while 
the  other  end  extends  below 
the  coil  and  connects  at  B. 
When  the  instrument  is 
joined  in  an  electric  circuit, 
the  current  is  conducted 
through  the  wire  of  the  coil. 
When  no  current  is  flowing, 
the  plane  of  the  coil  is  held 
by  the  suspending  wire  par- 
allel to  a  line  joining  the 
.two  poles  of  a  permanent 
magnet  N  and  S.  Since  a 
current  through  the  coil  de- 
velops a  magnetic  field  of  its 
own  at  right  angles  to  that  of 
the  magnet,  the  coil  will  turn 
until  the  magnetic  forces  are 
in  equilibrium  with  the  torsional  resistance  of  the  suspension  wire. 
The  deflections  are  read  by  the  movement  of  a  beam  of  light  reflected 
from  a  small  mirror  M  attached  to  the  coil.  For  small  angles  of  de- 
flection the  current  is  practically  proportional  to  the  angle  J  hence  the 
instrument  may  be  used  in  current  measurements.  The  instrument 
also  affords  a  very  sensitive  detector  of  currents  and  is,  on  this  account, 
indispensable  in  a  large  class  of  experiments. 
The  principle  involved  in  this  galvanometer 
is  employed  in  many  electrical  measuring 
instruments. 

425.  The  Ammeter.  —  An  instrument 
i  for  measuring  the  strength  of  an  electric  cur- 
rent is  called  an  ammeter,  or  amperemeter.  In 
most  ammeters  the  magnetic  effect  of  a  cur- 
rent is  employed.  The  instrument  may  con- 
sist simply  of  a  magnetic  needle,  which  shows 
by  the  amount  of  its  deflection  the  number 
of  amperes  of  current.  The  best  instruments,  however,  consist  of  a 
delicately  pivoted  coil  of  wire  A,  Fig.  344,  turning  between  the  poles 


FIG.  343.  —  The  d' Arson val  Galvanometer. 


FIG. 


344.  —  Section  of  au 
Ammeter. 


ELECTRICAL  MEASUREMENTS 


405 


of  a  strong  permanent  magnet  N  and  S  and  held  in  position  by  two 
hair  springs  a  and  &.  The  principle  involved  in  this  class  of  instru- 
ments will  be  recognized  at  once  as  that  of  the  d'Arsonval  galvanom- 
eter. When  a  current  is 
sent  through  the  coil,  the 
magnetic  field  developed  by 
the  current  causes  the  coil 
to  turn,  and  the  pointer  p 
moves  over  a  scale  which  is 
graduated  to  read  in  am- 
peres. When  the  current 
is  interrupted,  the  coil  is 
restored  to  its  initial  posi- 
tion by  the  springs  which 
also  serve  to  conduct  the 
current  into  and  out  of  the  FlG-  345.  — An  Ammeter, 

coil.     The  complete  instrument  is  shown  in  Fig.  345. 

426.  Electrical  Resistance.  —  It  was  observed  in  §  350 
that  substances  differ  in  respect  to  the   readiness  with 
which  they  transmit  an  electrical  charge ;  thus  bodies  are 
classed  as  good  or  poor  conductors.     The  opposition  that  a 
conductor  offers  tending  to  retard  the  transmission  of  elec- 
tricity is  called  electrical  resistance.     Hence,  with  a  given 
source  of  electricity,  as  a  Daniell  cell,  the  current  strength 
will  diminish  as  the  resistance  of  the  circuit  is  increased, 
and  will  rise  in  value  as  the  resistance  is  decreased. 

427.  Laws  of  Resistance.  —  Construct  a  frame  about  1  meter 
long  and  upon  it  stretch  4  wires  terminating  in  binding  posts.     Let 
No.  1  consist  of  1  meter  of  No.  30  (diameter  0.010  inch)  German  silver 
wire  ;  No.  2,  of  2  meters  of  No.  30  German  silver  wire ;  No.  3,  of  2  meters 
of  No.  28  (0.013  inch)  ;  and  No.  4,  about  20  meters  of  No.  30  copper 
wire.     Connect  wire  No.  1  in  series  with  one  or  two  Daniell  cells  and 
a  low  resistance  galvanometer,  and  read  the  deflection  of  the  needle. 
Replace  No.  1  by  No.  2,  and  the  deflection  will  be  found  to  be  less 
than  before,  thus  indicating  a  greater  resistance  for  the  greater  length. 
When  the  current  is  sent  through  No.  3,  which  is  a  larger  wire,  an 
increased  deflection  shows  a  decreased  resistance.     Finally,  when  the 
current  is  sent  through  wire  No.  4,  the  deflection  will  be  even  largei 


406         A   HIGH   SCHOOL  COURSE  ^N   PHYSICS 

than  for  No.  2,  which  is  of  the  same  size  and  only  one  tenth  as  long. 
Thus  copper  is  shown  to  be  more  than  10  times  as  good  a  conductor 
as  German  silver.  The  experiment  may  be  extended  to  other  wires 
of  various  substances. 

Accurate  measurements  verify  the  following  laws  : 

1.  The  resistance  of  a  conductor  of  uniform  size  and  com- 
position is  directly  proportional  to  its  length. 

2.  The  resistance  of  a  conductor  is  inversely  proportional 
to  its  cross-sectional .  area ;    or,  if  circular  inform,  to  the 
square  of  its  diameter. 

3.  The  resistance  of  a  conductor  depends  upon  the  nature 
of  the  substance  of  which  it  is  composed. 

For  example,  if  the  resistance  of  a  copper  wire  of  a 
certain  diameter  and  length  is  1  unit,  the  resistance  of  a 
wire  of  the  same  kind  and  size  and  twice  the  length  is 
2  units.  If,  now,  the  diameter  be  doubled,  the  resistance 
is  divided  by  22,  i.e.  reduced  to  2  units  -s-  4,  or  |  a  unit. 

428.  The  Unit  of  Resistance.  —  The  unit  of  resistance 
is  called  the  ohm  in  honor  of  Dr.  G.  S.  Ohm,1  a  German 
physicist.  The  ohm  is  the  amount  of  electrical  resistance 
offered  by  a  column  of  pure  mercury  106.3  cm.  in  height, 
of  uniform  cross  section,  and  having  a  mass  of  14.4521 
grams,  the  temperature  being  0°<7.  The  cross-sectional  area 
of  such  a  column  is  almost  exactly  one  square  millimeter. 
Since  such  a  column  of  mercury  would  be  inconvenient  to 
handle,  coils  of  wire  whose  resistances  have  been  carefully 
measured  and  recorded  are  used  in  practical  measure- 
ments. A  piece  of  No.  22  (0.025  inch)  copper  wire  60 
feet  in  length  has  approximately  one  ohm  of  resistance. 
One  meter  of  No.  30  German  silver  wire  has  a  resistance 
of  about  6.35  ohms.  A  convenient  unit  for  rough  work 
can  easily  be  made  by  winding  9  feet  and  5  inches  of 

1  See  portrait  facing  page  406. 


ANDRE    MARIE    AMPERE    (1775-1836) 


The  fame  of  Ampere  rests  mainly  on 
the  services  he  rendered  to  science  in  es- 
tablishing the  relation  between  electricity 
and  magnetism  and  in  developing  the  sci- 
ence of  electro-dynamics.  His  chief  experi- 
ments deal  with  the  magnetic  action  be- 
tween conductors  in  which  electric  currents 
are  flowing.  Ampere  was  also  the  first  to 
magnetize  needles  by  inserting  them  in  a 
helix  in  which  a  current  was  flowing. 

Ampere  was  born  at  Lyons,  France. 
During  the  French  Revolution  his  father 
was  beheaded,  an  event  which  for  years 
clouded  the  spirit  of  the  young  scientist. 
In  1805  he  became  professor  of  mathe- 
matics in  the  Polytechnic  School  in  Paris,  and  later  professor  of 
physics  in  the  College  of  France.  In  1823  he  published  his  mathe- 
matical classic  on  the  theory  of  magnetism.  The  practical  unit  of 
current  strength  is  named  in  his  honor. 


GEORGE    SIMON    OHM    (1787-1854) 

Following  closely  upon  the  experiments 
of  Ampere  on  the  magnetic  action  between 
currents  came  the  researches  of  Ohm,  a 
German,  whose  discoveries  concerned  the 
strength  of  an  electric  current. 

Ohm's  first  experiments  dealt  with  the 
conductivity  of  wires  of  different  metals. 
He  observed  the  deflections  of  a  magnetic 
needle  by  currents  from  a  given  source, 
but  flowing  through  different  conductors. 
By  changes  in  the  E.  M.  F.  in  the  circuits 
used,  he  obtained  results  which  led  him 

•pj 

to   the   well-known   expression    C  = . 

R  ~r  v 

He  then  investigated  cells  joined  in  paral- 
lel and  series,  and  published  his  results  in  1826.  During  the  following 
year  he  published  the  theoretic  deduction  of  the  law  bearing  his  name. 
This  law  is  one  of  the  most  fruitful  of  the  early  electrical  discoveries. 
Ohm  was  born  in  Erlangen,  where  he  was  educated.  His  ambi- 
tion to  become  a  university  professor  was  not  realized  until  1849, 
when  he  was  elected  to  a  professorship  at  Munich.  The  practical 
unit  of  resistance  is  called  the  ohm  in  his  honor. 


ELECTRICAL  MEASUREMENTS  407 

No.  30  copper  wire  on  a  spool  and  connecting  its  ends  to 
binding  posts. 

429.  Potential  Difference  —  the  Volt. — It  was   shown 
in  §  393  that  the  difference  of  potential  between  the  poles 
of  a  voltaic  cell  when  no  current  is  flowing  is  its  elec- 
tromotive force.      This  quantity  is  conceived    to   be    the 
cause  of  current  flow,  and  is  analogous  to  the  difference 
in  level  between  two  bodies  of  water,  to"  the  difference  in 
pressure  of   gases  compressed  in  two  reservoirs,  or  to  a 
difference  of  temperature  in  the  case  of  heat.     The  unit 
of  potential  difference  and,  consequently, of  E.  M.  F.,  is 
the  volt.      The  volt  is  that  difference  of  potential  between  the 
ends  of  a  wire  having  a  resistance  of  one  ohm  which  will 
produce  a  current  of  one  ampere.     The  E.  M.  F.  of  some 
of  the  voltaic  cells  in  ^common  use  is  given  in  the  follow- 
ing table : 

E.-M.F.  OF  CELLS 

Daniell,  1.1  volts.  Dry  cells,  1.5  volts. 

Leclanche,  1.5  volts.  Dichromate,  2.0  volts. 

A  Daniell  cell  would  then  cause  a  current  of  1.1  am- 
peres through  a  wire  having  a  resistance  of  1  ohm,  pro- 
vided there  were  no  other  resistance  in  the  circuit.  It  is 
evident,  however,  that  the  resistance  of  the  cell  itself 
must  always  be  considered  in  any  given  circuit. 

430.  The  Voltmeter.  —  In  order  that  an  instrument  may 
measure  the  E.  M.  F.  of  a  cell,  it  is  necessary  that  no  ap- 
preciable current  be  permitted  to  flow.     See  the  definition 
of  E.  M.  F.,  §  393.     In  order,  therefore,  that  a  galvanom- 
eter may  be  adapted  to  this  use,  it  must  contain  a  large 
amount   of  resistance.     Instruments  of  several  hundred 
ohms  are  made  whose  deflections  are  practically  propor-  \ 
tional   to   the    E.  M.  F.    of   cells   to  which  they  may  be 
attached.     When  such  instruments  are  graduated  to  read 
volts  they  are  called  voltmeters.     Figure   346  shows  the 


408 


A  HIGH   SCHOOL  COURSE  IN   PHYSICS 


manner   of  connecting  a  voltmeter  for  determining   the 
E.  M.  F.  of  the  cell  J5,  and  Fig.  347  shows  its  connection 


FIG.  346.  —  Illustrating  FIG.  347.  —  The  Voltmeter  Shows  the 

a  Use  of  the  Volt-  Fall  of  Potential  through  Coil  C  and 

meter.  the  Ammeter  Measures  the  Current 

•  Strength. 

for  giving  the  potential  difference  between  the  terminals 
of  a  coil  of  wire  placed  in  the  circuit.     One  well-known 

form     of     a    voltmeter     is 
shown   in   Fig.    348. 

431.  Ohm's  Law.  — Connect 
a  Daniell  cell  in  circuit  with  a 
galvanometer  or  ammeter  and  a 
resistance  box.  Make  the  resist- 
ance of  the  circuit  a  suitable 
amount  (say  200  ohms,  including 
the  galvanometer  resistance)  and 
measure  the  current  strength. 
Replace  the  Daniell  cell  by  a  new 
dry  cell  and  again  ascertain  the  current.  The  two  currents  will  be 
found  proportional  to  the  E.  M.  F.  of  the  cells  as  stated  in  the  table  in 
§  429.  Again  introduce  the  Daniell  cell  and  make  the  resistance  of 
the  entire  circuit  double  the  first  amount.  The  current  strength  will 
be  only  one  half  as  great.  Make  the  resistance  one  half  as  great  as  at 
first,  and  the  current  will  be  doubled. 

The  experiment  illustrates  the  law  first  published  by 
Dr.  G.  S.  Ohm  in  1827.  This  law  states  that  the  strength 
of  an  electric  current  is  directly  proportional  to  the  E.  M.  F. 


FIG.  348.  — The  Voltmeter. 


ELECTRICAL  MEASUREMENTS  409 

furnished  by  the  voltaic  cell  or  combination  of  cells,  and  in- 
versely proportional  to  the  total  resistance  of  the  circuit. 
The  ampere,  volt,  and  ohin  are  so  chosen  that  Ohm's  law 
may  be  written 


The  law  may  be  applied  to  any  portion  of  an  electric 
circuit,  in  which  case  it  becomes 

current  (amperes)  =  Potential  difference  (volts)  ^ 

resistance  (ohms) 

For  example,  the  current  produced  by  a  Daniell  cell 
(1.10  volts)  in  a  circuit  in  which  the  total  resistance 
(including  that  of  the  cell)  is  44  ohms  is  1.1-5-44,  or 
0.025  ampere.  It  is  clear  from  equation  (l)  that  if  any 
two  of  the  quantities  are  given,  the  third  can  easily  be 
computed. 

432.  Internal  Resistance.  —  The  current  that  any  cell 
can  produce  is  limited  by  the  resistance  which  the  current 
encounters  in  passing  through  the  liquid  of  the  cell.  This 
is  called  the  internal  resistance  of  the  circuit.  In  a  fresh  dry 
cell  the  resistance  is  a  fraction  of  an  ohm,  but  with  age  and 
use  it  increases  to  several  ohms.  That  of  a  Daniell  cell  varies 
from  about  one  to  three  or  four  ohms.  The  resistance  of  cells 
may  be  decreased  by  using  larger  plates  and  also  by  reducing 
the  distance  between  the  plates.  It  is  furthermore  depend- 
ent on  the  conductivity  of  the  liquid.  When  more  than 
one  cell  is  connected  in  a  circuit,  the  entire  internal  resist- 
ance depends  on  the  manner  in  which  they  are  joined 
together. 


410          A   HIGH   SCHOOL   COURSE   IN   PHYSICS 

433.   Cells  Connected  in  Series.  —  Join  each  of  three  Daniell 
cells  successively  to  a  voltmeter,  or  a  galvanometer  of  high  resistance 
(not  less  than  50  ohms),  and  record  the  deflections  produced.     They 
should  be  about  equal.     Now  connect  two 
of  the  cells  in  series,  as  shown  in  Fig.  349, 
and  lead  wires  to  the  instrument.     The  de- 
flection will  be  twice  as  great  as  for  one  cell, 
and  the  addition  of  the  third  cell  will  make 
the  current  three  times  as  strong.     The  de- 
flections under  these  conditions  are  propor- 
FIG.  349.  —  Two  Cells       tional  to  the  number  of  cells,  because  their 
Joined  in  Series.  electromotive  forces  are  added  together  when 

they  are  joined  in  this  manner. 

Cells  are  in  series  when  the  positive  pole  of  the  first  is 
joined  to  the  negative  pole  of  the  second,  the  positive  pole  of 
the  second  to  the  negative  pole  of  the  third*  and  so  on.  A 
study  of  the  figure  will  aid  greatly  in  making  the  method 
clear.  See  also  Fig.  311. 

Although  the  electromotive  forces  are  added  by  con- 
necting cells  in  series,  the  resistances  of  the  separate  cells 
also  add  together  and  thus  tend  to  reduce  the  current 
strength.  For  example,  four  similar  cells  in  series  will 
have  four  times  the  internal  resistance  of  one  cell. 

Since  the  strength  of  the  current  is  obtained  by  divid- 
ing the  E.  M.  F.  by  the  total  resistance  of  a  circuit, 


C  =       i2        .q        4          . 
R  +  rx  +  r2  +  r3  +  r4  +  etc.  ' 

where  Ev  E^  Ey  E^  etc.  are  the  electromotive  forces  of 
the  individual  cells,  rv  r2,  r3,  r4,  etc.  are  the  correspond- 
ing internal  resistances  and  R  is  the  external  resistance  of 
the  circuit.  If  we  have,  for  example,  five  similar  cells, 
the  equation  becomes 

,  and  for  n  cells,  C  =     nE    ,  (4) 

R  +  nr 


ELECTRICAL  MEASUREMENTS  411 

where  E  is  the  E.  M.  F.  and  r  the  resistance  of  a  single 
cell. 

434.   Cells  Connected  in  Parallel.  —  1.  With  the  apparatus 
used  in  the  experiment  of  the  preceding  section,  record  the  deflec- 
tion   produced    by    each 
cell  alone  and  then  con- 
nect the  positive  poles  of 
two  of  them  to  one  termi- 
nal of  the  instrument  and 
the  two  negative  poles  to 
the  other  terminal.     The 
deflection  will  be  the  same 
as    that    obtained    when 
only  one  cell  is  used.     If 
all  the  cells  are  joined  as 
shown  in  Fig.  350,  the  de-        FIG.  350.  —  Four  Cells  Joined  in  Parallel, 
flection  of  the  instrument  will  not  be  greatly  increased,  if  at  all. 

2.  Join  the  cells  while  connected  in  parallel  to  an  ammeter  or 
galvanometer  of  very  small  "resistance  and  record  the  reading.  Do 
the  same  with  the  cells  connected  in  series  and  also  with  only  one  cell. 
It  will  be  found  that  one  cell  alone  will  produce  about  as  much  cur- 
rent as  all  the  cells  when  joined  in  series,  but  a  much  stronger  current 
will  be  derived  from  them  when  they  are  in  parallel. 

The  experiments  show  that  the  parallel  arrangement  of 
cells  has  an  advantage  over  the  series  connection  when  the 
external  resistance  is  small.  In  fact,  when  the  external 
resistance  is  very  small,  as  it  is  in  some  cases,  the  series 
arrangement  of  cells  produces  practically  no  more  current 
than  a  single  cell.1  The  following  example  will  make  ihe 
matter  clear. 

The  current  obtained  from  a  single  cell  whose  E.  M.  F.  is  1.5  volts 
when  the  internal  resistance  is  3  ohms  and  the  external  resistance 

0.2  ohm  is,  by  equation  (1),  C  =     L5     ,  or  0.469  ampere. 

0.2  +  3 

1  When  cells  of  extremely  small  internal  resistance  are  used,  as  new 
dry  cells,  the  series  arrangement  is  the  better  under  nearly  all  circum- 
stances. A  poor  cell,  however,  having  a  large  resistance  may  prove  to  be 
more  of  a  hindrance  than  a  help. 


412         A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

Now  if  ten  such  cells  are  connected  in  series,  the  current  is,  by 

equation  (4),  C  =  10  x  1>5 ,  or  0.496  ampere,  which  is  only  slightly 
0.2  +  30 

larger  than  the  current  obtained  from  one  cell. 

Experiment  1  shows  that  the  E.  M.  F.  remains  un- 
changed when  cells  are  connected  in  parallel ;  hence, 
for  the  ten  cells  just  considered,  the  E.  M.  F.  is  just  the 
same  as  that  of  one  cell,  viz.  1.5  volts.  Again,  since  like 
plates  are  joined  together,  the  entire  combination  of  cells 
is  like  one  cell  having  plates  of  ten  times  the  area  of  those 
of  one  cell.  Therefore,  by  §  427,  the  internal  resistance  is 
only  one  tenth  as  much,  i.e.  3  -r-  10,  or  0.3  ohm.  The  equa- 
tion for  the  parallel  arrangement  becomes  C  = —-1 — -, 

or  3  amperes,  which  is  about  six  times  as  much  as  the 
series  arrangement  would  produce.  Hence  for  n  similar 
cells  joined  in  parallel,  Ohm's  law  is 

E.M.F.  of  one  cell  E 

~~  0  .  resistance  of  one  cell1        D  .  r 

-K-  1 : ~ ~ K-  ~\ 

number  of  cells  n 

EXERCISES 

The  pupil  should  represent  each  of  the  following  cases  diagram- 
matically. 

1.  In  the  discussion  of  the  two  methods  of  combining  cells,  for 
what  kind  of  circuits  is  the  series  arrangement  of  cells  shown  to  be 
suitable  ? 

2.  How  much  current  will  a  dry  cell  of  2  ohms  resistance  and 
1.43  volts  send  through  a  wire  of  25  ohms  resistance? 

3.  How  much  current  would  three  cells   similar  to  the  one  in 
Exer.  1  send  through  the  same  wire  (1)  when  joined  in  series  and 
(2)  in  parallel?  Ans.    (1)  0.13  ampere  ;  (2)  0.0557  ampere. 

4.  What  current  would  8  Dauiell  cells  of  which  the  E.  M.  F.  is 
1.08  volts  and  the  resistance  3  ohms  each  send  through  an  ammeter  of 
0.4  ohm,  when  joined  in  series?     What  would  be  the  current  from 
one  cell  alone  ? 


ELECTRICAL  MEASUREMENTS  413 

5.  What  would  be  the  current  produced  from  the  same  8  cells 
connected  in  parallel  and  using  the  same  ammeter  ? 

6.  Which  is  the  better  arrangement  of  4  Daniell  cells  (E.  M.  F. 
=  1.1  volts  and  r  =  2.5  ohms  each)  when  the  external  resistance  is 
6  ohms? 

7.  Show  that  a  small  Daniell  cell  will  give  practically  as  much 
current  as  a  large  one  through  1000  ohms  of  resistance,  the  resistance 
of  the  small  one  being  30  ohms  while  that  of  the  large  one  is  4  ohms. 

8.  If  a  galvanometer  gives  the  same  deflection  when  connected 
with  a  very  small  cell  as  it  does  when  connected  with  one  several 
times  as  large  but  having  the  same  E.  M.  F.,  is  the  resistance  of  the 
instrument  large  or  small  ? 

9.  A  dry  cell  whose  E.  M.  F.  is  1.5  volts  produces  a  current  of 
0.2  ampere  through  an  instrument  whose  resistance  is  7  ohms.     Find 
the  resistance  in  the  cell. 

10.  Which  will  produce  the  greater  effect  in  an  external  circuit  of 
5  ohms,  a  dry  cell  whose  resistance  is  3  ohms  or  a  copper-oxide  cell 
(E.  M.  F.  =  0.8  volt)  having  a  resistance  of  0.2  ohm? 

11.  What  current  will  be  derived  from  a  Daniell  cell  whose  resist- 
ance is  3  ohms  and  a  dry  cell  with  a  resistance  of  2  ohms  when  they 
are  joined  in  series  and  the  external  resistance  is  2.74  ohms? 

12.  What  would  be  the  value  of  the  current  in  the   preceding 
exercise  if  the  poles  of  the  Daniell  cell  were  set  in  opposition  to  those 
of  the  dry  cell?     Diagram  the  connections. 

13.  A  circuit  contains  4  dry  cells  (E.M.  F.  of  each  =  1.5  volts) 
joined  in  series.     Three  of  the  cells  are  known  to  have  a  resistance 
of  0.5  ohm  each,  and  the   external   resistance  is  10  ohms.     If  the 
current  is  0.15  ampere,  what  is  the  resistance  offered  by  the  fourth 
cell  ?    Would  the  current  be  increased  or  decreased  by  removing  this 
cell  from  the  circuit? 


2.    ELECTRICAL   ENERGY   AND    POWER 

435.  Energy  of  an  Electric  Current.  —  We  have  learned 
under  the  study  of  the  effects  of  electricity  (§§  411,  413, 
414)  that  the  energy  of  a  current  may  be  expended  in 
three  ways:  (1)  it  may  produce  mechanical  motion,  as  in 
the  electric  bell ;  (2)  it  may  produce  chemical  separation ; 
and  (3)  it  may  be  converted  into  heat  in  a  conductor. 


414         A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

Let  us  imagine  an  electric  circuit  as  shown  in  Fig.  351. 
Between  the  points  A  and  B  is  a  resistance  of  4  ohms. 
If  the  battery  produces  a  current  of  5  amperes,  for  ex- 
ample, the  potential  dif- 

£  4  Ohms,  5  Amperes  B 

^AAAAAAAA/V-+-V      ference  between  A  and  B 

20  Volte  \ 

will   be,    by    Ohm's    law 
(§-431),    4x5,     or     20 

volts- 

FIG.  351.  -Electrical  Energy  is  Trans-  Now  the  Work  done,  Or 

formed  into  Heat  between  A  and  B.  energy  expended,  by  the 
current  between  A  and  B  depends  on  three  factors, — 
(1)  the  potential  difference,  (2)  the  current  strength,  and 
(3)  the  time,  —  and  it  is  measured  by  their  product. 
Therefore  we  have  the  relation 

energy  =  potential  difference  x  current  strength  x  time. 

If  the  time  is  in  seconds,  the  potential  difference  in 
volts,  and  the  current  strength  in  amperes,  this  product 
gives  the  energy  in  terms  of  a  unit  called  the  joule, 
which  is  equal  to  10,000,000  ergs.  Hence 

volts  x  amperes  x  seconds  =  joules.  (6) 

In  the  electric  circuit  shown  in  Fig.  351  the  amount  of 
energy  expended  in  5  minutes  (300  seconds)  between  the 
points  A  and  B  is,  then,  by  equation  (6),  20  x  5  x  300,  or 
30,000  joules. 

436.  Power  of  an  Electric  Current.  —  Since  power  refers 
to  the  rate  at  which  work  is  done  or  energy  expended 
(§  57),  it  may  be  found  by  simply  dividing  the  total 
energy  expended  by  the  time.  In  an  electric  circuit, 
therefore,  the  power  is  measured  by  the  product  of  the 
potential  difference  and  the  current  strength ;  or, 

power  =  volts  x  amperes.  (?) 

It  is  plain  that  power  is  the  number  of  joules  per  second 
expended  by  the   current.     A  power   of   one   joule   per 


ELECTRICAL  MEASUREMENTS  415 

second  is  called  a  watt,  in  honor  of  James  Watt  (1736- 
1819)  of  Scotland.  One  horse  power  is  equal  to  746  watts. 
In  the  example  chosen  in  §  435,  the  power  of  the  current 
in  the  circuit  between  A  and  B  is  20  x  5,  or  100  watts  ; 
i.e.  if  the  energy  of  the  current  which  is  expended  be- 
tween the  points  A  and  B  could  be  converted  into  mechani- 
cal energy,  there  would  be  developed  ^J£  horse  power. 

437.  Quantity  of  Heat  Developed  by  a  Current.  —  When 
no  mechanical  or  chemical  work  is  done  by  a  current  of 
electricity,  the  energy  is  used  simply  in  overcoming  the 
resistance  of  the  conductor  and  is  converted  into  heat. 
Now  one  heat  unit  (a  calorie)  has  been  found  by  experi- 
ment to  be  equivalent  to  4.2  joules;  or,  one  joule  equals 

-— ,  or  0.24  calorie. 

If  we  express  the  potential  difference  between  two  points 
of  a  circuit  by  E  and  the  current  strength  by  (7,  the  num- 
ber of  joules  expended  in  t  seconds  is,  by  equation  (6), 

EOt  joules.     But  by  Ohm's  law  (§  431),  O=  ^;    whence 

.K 

E  =  OR.  Now  if  we  substitute  this  value  of  E,  we  obtain 
for  the  energy  expended  C2Rt,  which  represents  the  amount 
of  electrical  energy  that  is  converted  into  heat  in  a  resist- 
ance of  R  ohms  when  the  current  is  O  amperes  and  the 
time  t  seconds.  Reducing  joules  to  calories  gives 

Heat  =  0.24  C2Rt  calories.  (8) 

This  equation  represents  Joule's  law,  which  may  be 
stated  as  follows: 

The  heat  developed  in  a  conductor  by  an  electric  current  is 
proportional  to  the  square  of  the  current,  to  the  resistance  of 
the  conductor,  and  to  the  time  the  current  is  flowing. 

438.  Loss  of  Energy  in  Transmitting  Electricity.  —  The 
conversion  of  electrical  energy  into  heat  in  a  conductor 


416         A  HIGH   SCHOOL  COURSE  IN  PHYSICS 

through  which  it  flows  has  an  important  commercial  bear- 
ing on  the  transmission  of  electric  power  over  long  lines. 
For  example,  if  the  resistance  of  the  wires  leading  from  a 
distant  source  of  electrical  power  to  a  group  of  lamps  is 
only  2  ohms,  the  waste  of  power  in  transmitting  a  current 
of  10  amperes  is  102  x  2,  or  200  watts.  Now  10  amperes 
at  110  volts  would  operate  20  lamps,  each  of  which  con- 
sumes 55  watts.  The  total  power  required  would  be, 
therefore,  20  x  55  +  200,  or  1300  watts.  The  loss  in 
transmission  is  therefore  -fy  of  the  total  power  produced. 
Now  if  the  number  of  lamps  is  increased  to  100,  the  lamp 
consumption  is  5500  watts,  while  the  line  loss  (since  the 
current  is  now  50  amperes)  is  502  x  2,  or  5000  watts. 
Hence,  of  the  10,500  watts  which  must  be  produced  at  the 
power  station,  5000  watts,  or  47.6  per  cent,  are  lost  by 
being  converted  into  heat  in  the  line. 

The  loss  of  energy  in  long  lines  of  wire  can  be  reduced 
by  constructing  the  line  of  larger  wire  and  thus  decreasing 
the  resistance  factor,  but  in  many  cases  this  method  is  im- 
practicable from  a  commercial  standpoint.  However,  by 
using  modern  devices  involving  principles  that  we  shall 
study  later,  economical  transmission  of  power  over  long 
distances  is  rendered  possible. 

EXERCISES 

1.  Compute  the  number  of  joules  transmitted  by  a  current  of  10 
amperes  maintained  for  20  minutes  at  a  potential  difference  of  110 
volts. 

2.  Compute  the  heat  loss  per  hour  in  an  electric  line  of  3  ohms 
resistance  when  the  current  is  5  amperes.     What  is  the  result  if  the 
current  is  10  amperes  ? 

3.  If  the  power  required  for  an  incandescent  lamp  is  60  watts,  what 
is  the  consumption  of  50  lamps  measured  in  terms  of  the  horse  power? 

4.  A  piece  of  platinum  wire  is  heated  by  a  current  of  2  amperes, 
and  the  potential  difference  between  its  ends  is  8  volts.     Compute  the 
heat  developed  per  minute. 


ELECTRICAL   MEASUREMENTS  417 


3.  COMPUTATION  AND  MEASUREMENT  OF  RESISTANCES 

439.  Resistances  Computed. — It  is  customary  to  com- 
pute the  resistance  of  a  wire  of  given  material,  length, 
and  size  from  the  known  resistance  of  a  wire  of  that  kind 
which  is  1  foot  in  length  and  0.001  inch  (called  1  mil) 
in  diameter.  The  following  table  gives  this  value  in 
ohms  for  some  of  the  common  metals. 

OHMS  OF  RESISTANCE  IN  WIRES  1  FOOT  LONG  AND  0.001  INCH  IN 

DIAMETER 

Silver 9.5  Platinum 80 

Copper 10.2  German  silver    ....     180 

Iron 61.5  Mercury 570 

According  to  the  laws  of  resistance  stated  in  §  427,  the 
resistance  of  a  wire  can  be  calculated  by  simply  multiplying 
the  number  given  in  the  table  by  the  length  of  the  wire  in  feet 
and  dividing  by  the  square  of  the  diameter,  which  must  first 
be  reduced  to  thousandths  of  an  inch.  For  example,  the 
resistance  of  a  mile  of  iron  wire  0.08  inch  in  diameter  is 
61.5  ohms  x  5280  -5-  802,  or  50.7  ohms. 

EXERCISES 

1.  What  is  the  resistance  of  each  of  the  wires  given  in  the  follow- 
ing table? 

KIND  OF  WIRE  NUMBER  DIAMETER  LENGTH 

Copper 8  0.128  inch  1  mile 

Copper 22  0.025  inch  500  feet 

Copper 36  0.005  inch  40  feet 

Copper 40  0.003  inch  25  feet 

Iron 9  0.114  inch  1  mile 

Iron 14  0.064  inch  25  feet 

Iron 10  0.102  inch  5000  feet 

German  silver 20  0.032  inch  35  feet 

German  silver 30  0.010  inch  10  meters 

Platinum      .     .          ....  24  0.020  inch  2  feet 

"28 


418          A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

2.  How  long  must  a  No.  36  copper  wire  be  to  offer  a  resistance  of 
one  ohm  ? 

3.  Find  the  diameter  of  a  copper  wire  that  has  a  resistance  of  10 
ohms  per  thousand  feet. 

4.  A  No.  22  wire  1000  ft.  long  offers  a  resistance  of  285  ohms. 
Of  what  material  mentioned  in  §  439  might  the  wire  be  made  ? 

440.  Resistance  Boxes.  —  Coils  of  wire  whose  resistances 

have  been  carefully  adjusted  and 
marked  are  incased  in  boxes  and 
sold  by  electrical  supply  houses. 
Such  a  box  is  shown  in  Fig.  352. 
The  ends  of  each  coil  are  sol- 
dered to  brass  blocks  A,  B,  C, 
FIG.  352.  — A  Resistance  BOX.  'etc.,  Fig.  353,  between  which 

brass  plugs  may  be  inserted.     The  blocks  are  mounted  on 

an  ebonite  or  hardwood  plate  which  forms  the  cover  of 

the  box.     When  the  plugs  are  all  in  place, 

no  resistance  is  encountered  by  an  electric 

current  in  passing  from  one  block  to  the 

next ;  but  when  a  plug  is  withdrawn,  the 

current  must  pass  through  the  correspond-  FIG.  353.  —  Showing 

ing  coil  whose  resistance  is  marked  on 

the  top  of  the  box.     Thus,  by  removing 

plugs,  any  resistance  from  the  smallest  up  to  the  sum  of 

all  the  resistances  in  the  box  can  be  obtained. 

441.  Resistance  Measured  by  Substitution. — Place  a  coil 
of  wire  whose  resistance  is  to  be  determined  in  circuit  with  a  Daniell 
cell  and  a  galvanometer  and  read  the  deflection  produced  by  the  current. 
Remove  the  coil  from  the  circuit  and  insert  a  resistance  box.     With- 
draw plugs  from  the  box  until  the  deflection  of  the  galvanometer  is 
the  same  as  before.     The  sum  of  the  resistances  corresponding  to  the 
plugs  that  have  been  removed  gives  the  resistance  of  the  coil.     Why  ? 

442.  Resistances  Measured  by  Voltmeter  and  Ammeter.  — 

The  coil  of  wire  C,  Fig.  347,  whose  resistance  is  to  be  found,  is  placed 
in  a  circuit  with  the  cell  B  and  an  ammeter.     A  voltmeter  of  high 


ELECTRICAL   MEASUREMENTS  419 

resistance  is  joined  to  the  terminals  of  the   coil  C  to  indicate   the 

potential  difference  at  those   points.      Since  by   Ohm's  law  (§  431) 

the  current  through  C  equals 

the  potential  difference  in  volts 

divided   by  the   resistance  in 

ohms  (Eq.  2,  §  431),  the  value 

of  C  can  be  found  by  dividing 

the  reading  of  the  voltmeter 

by  that  of  the  ammeter.     It  is 

essential  that  the  resistance  of 

the  voltmeter  be  so  large  that 

practically  none  of  the  current        FIG.  354.  —  Diagram  of  Wheatstone's 

can  pass  through  it.  Bridge. 

443.  The  Wheatstone  Bridge.  —When  the  current  from  a  cell 
J3,  Fig.  354,  passes  through  the  point  A,  it  divides  into  two  parts,  the 
portion  C  flowing  through  the  resistance  R  ohms,  and  the  portion  C' 
through  the  resistance  R'  ohms  in  the  straight  uniform  wire.  AD. 
Now  a  sensitive  galvanometer  G  joining  points  E  and  F  will  be 
deflected  unless  E  and  F  do  not  differ  in  potential.  The  point  F  which 
has  the  same  potential  as  E  is  found  by  moving  the  contact  along 
the  wire  AD  until  no  deflection  can  be  observed. 

Under  these  conditions  the  difference  of  potential  between  A  and 
E  must  equal  that  between  A  and  F.  By  Ohm's  law  (§  431), 

fall  of  potential  -4-  resistance  =  current. 
Hence  fall  of  potential  =  current  x  resistance', 

whence  CR  =  C'R'.  (l) 

Again,  the  same  current  flows  through  X  as  through  R,  since  the 
galvanometer  is  not  deflected  ;  and  the  same  current  flows  through 
R"  as  though  R'  for  the  same  reason.  Now,  since  the  difference  of 
potential  between  E  and  D  is  the  same  as  that  between  F  and  D,  we 

have  CX  =  C'R".  (2) 

Dividing  (2)  by  (1)  we  have 


Now  the  resistances  R'  and  R"  are  proportional  to  the  lengths  of 
the  wire  AF  and  FD  (§  427).     Therefore, 

X  =  R.  (3) 


420          A  HIGH   SCHOOL  COURSH   IN    PHYSICS 

Four  resistances  combined  in  this  manner  constitute  Wheatstone's 
Bridge.  It  is  obvious  that  if  the  resistance  R  and  the  lengths  of  the 
wire  AF  and  FD  are  known,  the  value  of  the  unknown  resistance  X 
can  be  calculated.  The  point  F  must  be  found  experimentally  as 
described  above. 

444.  Conductors  in  Series  and  Parallel.  —  Conductors 
are  said  to  be  joined  in  series  when  they  are  connected  in 
>>i  succession  as  shown  in 

-""W^^-*-^^  Fig.  355.     In  this  con- 

FIG.  355. -Conductors  Joined  in  Series.     dition  the  entire  current 

passes  through  each  conductor.  The  combined  resistance 
between  A  and  B  is  the  sum  of  the  resistances  of  the  sev- 
eral parts. 

The  case  is  very  different  when  the  conductors  are 
joined  as  shown  in  Fig.  356.  Resistances  combined  in 
this  manner  are  said  to 
be  connected  in  parallel. 
The  most  important  com- 
bination of  this  kind  is 
that  of  two  wires  thus  con- 
nected. The  current  from 
the  cell  O  will  obviously 

divide    into    tWO    parts    at  FIG.  356. -Conductors  Joined 

point  A  which  reunite  at  in  Parallel. 

B.  When  the  two  resistances  are  equal,  the  two  parts  of 
the  current  are,  of  course,  equal.  But  if  the  resistances 
are,  for  example,  3  and  7  ohms  respectively,  and  the  poten- 
tial difference  between  A  and  B  common  to  both  branches 
is  1  volt,  the  current  through  the  upper  branch  is  J  am- 
pere and  that  through  the  lower  one  ^  ampere.  Hence, 
while  the  resistances  are  as  3  is  to  7,  the  currents  are  as  7 
is  to  3.  The  currents  from  the  cell,  therefore,  in  a  two- 
branched  circuit  are  inversely  proportional  to  the  correspond- 
ing resistances. 


ELECTRICAL   MEASUREMENTS  421 

Again,    the   total   current   flowing   between  A   and  B 
is  ^  +  ijr  ampere.     If,  now,  we  let   R   be    the   combined 

resistance  between  A  and  B,  the  total  current  is  —  ampere. 

R 

Therefore  issl+i,  (4)    ^ 

3x7  V 

whence  R  = -,  or  2.1  ohms.     It   is   clear   from   the 

3  +  7 

example  chosen,  that  the  combined  resistance  offered  by  the 
two  branches  of  a  divided  electric  circuit  is  the  product  of 
the  two  resistances  divided  by  their  sum;  or,  expressed  as 

an  equation,  R=?12<l2.  (5) 


445.  Shunts. — A  shunt  is  a  conductor  which  is  con- 
nected in  an  electric  circuit  parallel  to  another  conductor. 
It  may  be  likened  to  a  side-track,  or  to  a  by-pass  that  is 
frequently  used  in  the  case  of 
the  conduction  of  water  or  gas 
through  pipes.  The  term  is 
applied  to  a  resistance  coil  $, 
Fig.  357,  which  is  joined  in  a 
circuit  parallel  to  a  galvanom- 
eter Gr  or  some  other  instru- 

C\ ' 

ment.  In  this  case  it  is  used  FlG  357. -illustrating  the  Use  of 
to  reduce  the  flow  of  electricity  a  Shunt, 

through  the  instrument;  for,  as  shown  in  §  444,  the  cur- 
rent divides  into  parts  that  are  inversely  as  the  two  resist- 
ances. Suppose,  for  example,  the  resistance  of  Q-  is  45 
ohms  and  that  of  S  is  5  ohms.  Then  ^  of  the  entire  cur- 
rent passes  through  the  galvanometer  .and  £jj-  of  it  through 
the  shunt.  Thus  it  is  clear  that  when  a  shunt  contains  ^  as 
much  resistance  as  a  galvanometer,  yL-  of  the  total  current 
passes  through  the  galvanometer,  and  ^  through  the 


422          A   HIGH   SCHOOL  COURSE   IN   PHYSICS 

shunt.     In  the  same  manner  the  division  of  a  current  in 
any  given  case  may  be  computed. 

EXERCISES 

The  pupil  should  diagram  the  conditions  expressed  in  each  of  the 
following  exercises. 

1.  A  current  of  4  amperes  divides  and  passes  through  two  parallel 
coils  of  wire,  one  of  5  ohms  and  the  other  of  8  ohms.     What  is  the 
current  that  goes  through  each  branch? 

Ans.   2.461  amperes  and  1.539  amperes. 

2.  Find  the  combined  resistance  offered  by  the  two  parallel  con- 
ductors in  Exer.  1. 

3.  Two  instruments  of  3  and  4  ohrns  respectively  are  connected 
parallel  between  the  poles  of  a  dry  cell  (E.M.  F.  1.5  volts)  whose  re- 
sistance is  1  ohm.     Find  (1)  the  external  resistance  of  the  circuit, 
(2)  the  current  strength  of   the  cell,  and  (3)  the  current  flowing 
through  each  branch. 

Ans.   (1)  1.714  ohms;    (2)  0.553   ampere;    (3)  0.316   ampere, 
0.237  ampere. 

4.  Two  electro-magnets  of  4  and  12  ohms  respectively  are  connected 
in  parallel  to  each  other  and  then  placed  in  a  circuit  containing  a 
coil  of  3  ohms  and  a  battery  of  10  ohms.     Find  (1)  the  total  resist- 
ance of  the  circuit,  and  (2)  the  strength  of  the  current  in  each  part 
of  the  circuit,  the  E.  M.  F.  being  4.8  volts. 

Ans.   (1)  16  ohms;  (2)  0.225  ampere  and  0.075  ampere. 

5.  A  galvanometer  of  30  ohms  has  a  shunt  of  30  ohms.     When 
connected  in  an  electric  circuit,  what  part  of  the  whole  current  will 
pass  through   the   instrument?      By  what   number  must  the   cur- 
rent measured  by  the  galvanometer  be  multiplied  to  give  the  entire 
current  ? 

6.  If  the  resistance  of  the  shunt  in  Exer.  5  is  reduced  to  20  ohms, 
what  part  of  the  total  current  will  the  galvanometer  carry,  and  what 
will  be  the  multiplier?  Ans.    The  multiplier  will  be  2.5. 

7.  If  a  galvanometer  has  300  ohms  of  resistance,  what  must  be  the 
resistance  of  a  shunt  so  that  only  one  tenth  of  the  entire  current  will 
pass  through  the  wire  of  the  instrument?  Ans.   33£  ohms. 

SUMMARY 

1.    The  unit  of  current  strength  is  the  ampere.     It  is 
the  current  that  will  deposit  0.001118  g.  of   silver   per 


ELECTRICAL  MEASUREMENTS  423 

second.     Current  strength  is   measured   by  an  ammeter 
(§  422). 

2.  The    electrical    resistance    of   cylindrical   wires   of 
uniform  size  and  composition  is  directly  proportional  to 
the  length  and  inversely  proportional  to  the  square  of  the 
diameter.     It  also  varies  with  the  nature  of  the  substance 
of  which  it  is  composed.     The  unit  of  resistance  is  the 
ohm  (§§  427  and  428). 

3.  The   unit  of  potential  difference  is  the  volt.     It  is 
the  potential  difference  required  to  force  a  current  of  an 
ampere  through  a  resistance  of  one  ohm.     Potential  dif- 
ferences are  measured  by  the  voltmeter  (§§  429  and  430). 

4.  Current,  potential  difference,  and  resistance  are  re- 

F 

lated  mathematically  as  shown  by  the  equation  C=  —  ,  or 

R 

amperes  =  volts  -^  ohms.  -    This  relation  is  known  as  Ohm's 
Law  (§  431). 

5.  The   resistance   of   circuits    includes    internal    and 
external  resistances  ;  the  former  is  the  resistance  offered 
by  the  cells,  the  latter,  that  of  the  remaining  portion  of 
the  circuit  (§  432). 

6.  For   n   similar  cells  joined  in  series  the  current  is 

0=     nE      (§  433). 
R  +  nr  ^ 

7.  For  n  similar  cells  joined  in  parallel  the  current  is 

0=  —  —  (§  484). 


8.  The  energy  expended  in  any  part  of  a  circuit  is  the 
product  of  the  potential  difference,  current  strength,  and 
time,  or 

energy  =  ECt  joules  (§  435). 


424          A  HIGH   SCHOOL   COURSE   IN   PHYSICS 

9.    The  power,  or  rate  at  which  the  current  is  working, 
is  power  =  EC  watts  (§  436). 

10.  The  quantity  of  heat  developed  by  a  current  of 
O  amperes  in  a  resistance  R  ohms  in  t  seconds  is 

heat=  0.24  C*Rt  calories  (§  437). 

11.  The  resistance  of  a  cylindrical  wire  may  be  com- 
puted by  multiplying  the  resistance  of  one  foot  of  it  one 
mil  in  diameter  by  the  whole  length  and  dividing  by  the 
square  of  the  diameter  in  mils  (§  439). 

12.  The  combined  resistance  of  conductors   joined  in 
series  is  their  sum.     The  combined  resistance  of  two  con- 
ductors joined  in  parallel  is 


CHAPTER   XX 

ELECTRO-MAGNETIC   INDUCTION 

1.    INDUCED  CURRENTS   OF   ELECTRICITY 

446.  Currents  Induced  by  Magnetism.  —  Oersted's  dis- 
covery of  the  effect  of  a  current  of  electricity  upon  a  mag- 
netic needle  (§  400)  in  1819  led  to  the  invention  of  the 
electro-magnet  by  Sturgeon  in  1825,  and  induced  many 
experimenters  to  seek  for  a  method  of  producing  an  elec- 
tric current  by  means  of  a  magnet.  Two  physicists, 
Joseph  Henry1  in  America  and  Michael  Faraday1  in  Eng- 
land, independently  discovered  the  process  for  doing  this 
about  1831. 

Connect  the  ends  of  a  coil  of  insulated  wire  C,  Fig.  358,  consisting 
of  a  large  number  of  turns,  directly  to  the  termi- 
nals of  a  sensitive  d'Arsonval  galvanometer.  Ar- 
range a  large  horseshoe  magnet  with  its  poles 
upward  as  shown.  Now  move  the  coil  down  quickly 
into  the  magnetic  field.  The  galvanometer  will  re- 
veal the  presence  of  a  current  of  electricity,  but  the 
index  will  go  back  to  zero  as  soon  as  the  coil  stops  mov- 
ing. If  the  coil  be  now  removed  from  the  magnetic 
field,  the  galvanometer  will  show  that  a  current  is 
produced  in  the  opposite  direction.  Repeat  the  ex- 
periment, but  move  the  coil  more  slowly.  Turn  the 
coil  over  and  repeat  the  experiment.  Each  deflec- 
tion will  be  in  a  direction  opposite  to  the  corres- 
ponding one  produced  at  first.  FIG.  358.  —  Induc- 
ing an  Electric 

The  experiment  shows  clearly  the  produc-       Current. 
tion  of  an  electric  current  without  the  aid  of  a  voltaic  cell. 


1  See  portraits  facing  page  426.      See  also  Maxwell,  facing  page  432. 

425 


426          A  HIGH   SCHOOL   COURSE   IN   PHYSICS 

Such  a  current  is  called  an  induced  current.  The  experi- 
ment shows  also  that  the  induced  current  flows  only  while 
the  coil  is  moving  in  the  magnetic  field,  i.e.  only  while  the 
number  of  lines  of  force  through  the  coil  is  changing. 
Since  a  current  is  always  due  to  an  E.  M.  F.,  it  is  obvious 
that  the  movement  of  a  coil  of  wire  in  the  magnetic  field 
develops  such  an  electromotive  force  in  the  wire.  This 
is  called  an  induced  electromotive  force.  Furthermore;  the 
slower  the  movements,  the  less  rapid  the  change,  and  the 
smaller  the  induced  E.  M.  F.  The  following  general  laws 
may  be  stated : 

1.  A  change  in  the  number  of  magnetic  lines  of  force 
threading  through  a  coil  (or  a  single  loop)  of  wire  induces 
an  electromotive  force  in  that  coil. 

2.  The  induced  electromotive  force  is  proportional  to  the 
rate  at  which  the  number  of  lines  of  force  is  changed. 

447.  Special  Cases  of  Current  Induction.  —  According  to 
the  laws  of  induction  given  in  the  preceding  section,  an 
induced  electromotive  force  may  be  brought  about  either 
by  an  increase  or  by  a  decrease  in  the  number  of  magnetic 
lines  through  a  coil  of  wire.  This  effect  may  be  produced 
in  several  different  ways,  as  the  following  experiments 
will  show. 

1.  Connect  the  ends  of  a  coil  of  wire  with  a  sensitive  galvanometer 
and  thrust  the  N-pole  of  a  magnet  into  the  coil.     A  deflection  will  be 
produced.     On  pulling  the  same  pole  out  of  the  coil,  a  deflection  in 
the  opposite  direction  results.     Turn  the  coil  over  and  repeat  the 
operations.     Every  effect  is  just  the  reverse  of  the  corresponding  one 
produced  at  first.     The  experiment  should  be  repeated,  using  the  S- 
pole  in  the  same  manner. 

2.  Connect  a  coil  of  wire,  Fig.  359,  to  the  poles  of  a  voltaic  cell 
and  thrust  this  coil  into  the  coil  used  in  Experiment  1.     In  every 
operation  similar  to  those  performed  above,  the  effect  is  the  same  as 
that  produced  by  using  a  magnet.     Repeat  with  a  soft  iron  core  within 
the  smaller  coil. 


MICHAEL    FARADAY    (1791-1867) 

Faraday  was  one  of  the  most  distin- 
guished chemists  and  physicists  of  the 
nineteenth  century.  He  was  the  son  of 
a  blacksmith  at  Newington,  near  London, 
England,  and  became  a  bookbinder  in  1804. 
Hearing  by  chance  some  lectures  on  chem- 
istry by  Sir  Humphry  Davy  of  the  Royal 
Institution,  he  became  interested  in  the 
subject,  and  in  1813  was  appointed  assist- 
ant in  Davy's  laboratory.  In  1825  he  be- 
came director  of  this  laboratory,  and  in 
1833  was  made  professor  of  chemistry  in 
the  Royal  Institution  for  life.  He  died  at 
Hampton  Court  in  1867. 

Faraday's  achievements  in  the  domain 
of  chemistry  are  largely  overshadowed  by  his  numerous  and  brilliant 
discoveries  in  electricity,  of  which  the  most  far-reaching  was  that 
of  the  induction  of  electric  currents  by  magnets,  made  in  1831.  This 
subject  has  led  to  results  of  tremendous  value  in  the  commercial 
applications  of  electricity. 

His  further  discoveries  d_cal  with  the  capacity  of  condensers,  the 
laws  of  electrolysis,  the  rotation  of  the  plane  of  polarized  light  by 
a  magnetic  field,  and  diamagnetism. 

Faraday  was  a  prolific  writer  and  a  popular  lecturer  on  scien- 
tific subjects.  He  is  best  known  by  his  Experimental  Researches 
in  Chemistry  and  Physics. 


JOSEPH    HENRY    (1797-1878) 


After  the  time  of  Franklin,  Henry  was 
the  first  in  the  United  States  to  make  orig- 
inal researches  in  the  subject  of  electricity. 
After  the  invention  of  a  practical  electro- 
magnet by  Sturgeon  in  1825,  Henry  devel- 
oped a  magnet,  in  the  construction  of  which 
he  employed  many  turns  of  copper  wire  in- 
sulated with  silk.  The  magnet  was  capable 
of  supporting  over  fifty  times  its  own  weight. 
It  is  believed  on  good  evidence  by  many 
physicists  that  the  discovery  of  electro- 
magnetic induction  was  made  by  Henry  at 
Albany,  NTew  York,  in  1830,  although  he  did 
not  publish  his  results  until  1832,  a  year  later 
than  the  publication  of  Faraday's  results. 


ELECTRO-MAGNETIC  INDUCTION 


427 


3.  Open  the  battery  circuit  used  in  Experiment  2,  insert  a  key,  and 
place  one  coil  within  the  other.     Complete  the  battery  circuit  by  press- 
ing the  key.     A  deflection 

of  the  galvanometer  will 
indicate  the  presence  of  an 
induced  current  which  at 
once  dies  away.  Open  the 
key,  and  an  induced  cur- 
rent in  the  opposite  direc-  Q 
tion  will  be  obtained. 

4.  If  the  galvanometer 

is  sufficiently  sensitive,  aim-  FlG"  m -InducinS  a  Current  ^  a  Cl»™nt. 
ply  turning  the  coil  with  which  it  is  connected  in  the  earth's  magnetic 
field  will  suffice  to  produce  an  appreciable  current. 

It  is  to  be  observed  that  in  each  of  the  cases  chosen  in 
the  experiments,  a  change  in  the  number  of  magnetic  lines 
through  the  coil  is  brought  about  in  the  process.  In  Ex- 
periment 1,  the  magnet-  carries  its  lines  of  force  with  it 
when  it  is  thrust  into  the  coil ;  in  Experiment  2,  the  mag- 
netic lines  of  force  set  up  by  the  battery  current  in  one 
coil  are  carried  through  the  second  coil  when  the  former 
is  thrust  into  the  latter ;  in  Experiment  3,  the  change  in 
the  number  of  lines  is  produced  by  alternately  making 
and  breaking  the  circuit  which  contains  the  battery.  In 

every  case  the  induced  current 
is  set  up  by  magnetic  action,  i.e. 
'ni  either  by  an  increase  or  a  decrease 
in  the  number  of  magnetic  lines 
through  the  coil  connected  with 
the  galvanometer. 


Induced 


448.  Direction  of  the  Induced 
Current.  —  Lenz's   Law.  —  Connect 
a  small  piece  of  zinc  with  one  terminal 
of  a  sensitive  galvanometer  and  a  piece 
FIG.  360.  —  Showing  the  Direc-  of  c°PPer  with  the  other.     Hold  the  two 
tion  of  the  Induced  Current,      metals  in  the  fingers  and  observe   the 


428          A   HIGH  SCHOOL  COURSE   IN    PHYSIC'S 

deflection  of  the  galvanometer.  The  moisture  of  the  hand  is  sufficient 
to  establish  a  current  from  the  copper  to  the  zinc  through  the  instru- 
ment. Now  repeat  the  experiment  of  §  446,  and  from  the  direction 
in  which  the  galvanometer  is  deflected,  determine  the  direction  of  the 
induced  current  in  the  wire. 

While  the  coil  is  moving  into  the  magnetic  field  placed 
as  shown  in  Fig.  360,  the  current  in  the  wire  will  be 
found  to  have  the  direction  shown  by  the  arrows.  Thus 
the  induced  current  tends  to  set  up  a  magnetic  field  of  its 
own  whose  lines  of  force  are  opposite  in  direction  to  those 
of  the  magnet.  This  fact  is  easily  verified  by  applying  the 
rule  given  in  §  408.  See  Fig. 
3610  Hence,  increasing  the  mag- 
netic lines  through  the  coil  produces  a 
current  which  tends  to  oppose  that  in- 
crease. On  the  other  hand,  moving 
the  coil  away  from  the  magnet  in- 
duces in  the  coil  a  current  in  the 
direction  opposite  to  the  former  cur- 
FIG.  361.  — Applying  rent.  Hence,  a  decrease  in  the  num- 

Lenz's  Law.  7          /.7.          ,7  7    ,7  .7 

per  of  lines  through  the  coil  sets  up  a 

current  that  tends  to  prevent  that  decrease  in  number.  In 
fact,  every  case  of  current  induction  may  be  shown  to  obey 
the  following  law: 

The  direction  of  an  induced  current  is  such  as  to  produce 
a  magnetic  field  that  will  tend  to  prevent  a  change  in  the 
number  of  magnetic  lines  of  force  through  the  coil. 

This  is  known  as  Lenz's  Law  and  may  be  viewed  as  an 
application  of  the  more  general  law  of  the  Conservation  of 
Energy  (§  64).  From  this  law  we  know  that  neither  elec- 
trical nor  any  other  form  of  energy  can  be  derived  unless 
an  equivalent  amount  of  work  be  performed.  In  this 
case  the  work  is  done  when  the  coil  is  forced  into  the 


ELECTRO-MAGNETIC   INDUCTION 


429 


FIG.  362.  — Diagram  Showing  the  Parts  of 
an  Induction  Coil. 


magnetic  field  in  opposition  to  the  repulsion  between  the 
field  of  the  magnet  and  that  of  the  induced  current. 

449.  The  Induction  Coil.  —  One  of  the  most  important 
applications  of  the  principles  of  electro-magnetic  induction 
is  found  in  the  induction 
coil.  The  instrument 
consists  of  a  so-called 
primary  coil  of  coarse 
wire  J.,  Fig.  362,  wound 
on  a  core  of  soft  iron,  a 
secondary  coil  of  several 
thousand  turns  of  very 
fine  insulated  wire  S, 
and  a  current  interrupter 
I.  The  terminals  of  the 
secondary  coil  are  at  p 
and  q. 

When  the  current  from  a  battery  B  flows  through  the 
primary  coil,  it  magnetizes  the  iron  core.  The  iron  ham- 
mer a  is  then  drawn  toward  the  core  and  away  from  the 
screw  b.  This  operation  breaks  the  primary  circuit  at  the 
screw  point  d.  The  interruption  of  the  current  at  d  causes 
the  core  to  lose  its  magnetism  and  release  the  hammer  a, 
which  is  restored  to  its  original  position  by  the  spring  to 
which  it  is  attached.  The  contact  at  d  is  thus  made  again, 
and  all  the  operations  are  repeated.  It  is  clear  that  most 
of  the  lines  of  force  set  up  by  the  battery  current  in  the 
primary  coil  pass  through  each  turn  of  the  secondary. 
Hence,  when  the  current  is  interrupted  and  the  lines  of 
force  decrease,  an  E.  M.F.  is  induced  in  the  secondary  coil. 
Even  with  small  coils,  the  potential  difference  between  p 
and  q  is  often  sufficient  to  cause  a  spark  to  pass  across  a 
short  gap  from  one  to  the  other.  Again,  when  the  primary 
circuit  is  made  at  d,  the  resulting  increase  in  the  number 


430         A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

of  lines  of  force  induces  a  contrary  E.  M.  F.  in  the  second- 
ary, but  this  is  of  a  much  smaller  intensity  than  the  former 
induced  E.  M.  F.  Lenz's  law  applied  to  the  induction 
coil  shows  the  following  statements  to  hold  true  : 

1.  When  a  current  is  started  in  the  primary  coil,  a  momen- 
tary current  is  induced  in  the  secondary  in  the  opposite 
direction. 

2.  When  the  current  in  the  primary  coil  is  interrupted,  a 
momentary  current  is  induced  in  the  secondary  in  the  same 
direction  as  that  in  the  primary. 

The  condenser  is  introduced  as  an  accessory  part  of  the 
induction  coil  in  order  to  increase  the  rate  of  demagnetiza- 
tion of  the  iron  core.  By  its  aid  a  very  quick  interruption 

of  the  current  in  the 
primary  coil  is  ac- 
complished, and 
consequently  the  in- 
duced E.  M.  F.  is 
proportionately  in- 

FIG.  363.  — An  Induction  Coil.  ,         r™ 

creased.      The    use 

of  an  extremely  large  number  of  turns  of  wire  in  the 
secondary  coil  makes  it  possible  to  secure  a  sufficiently 
large  potential  difference  between  p  and  q  to  produce 
sparks  many  inches  in  length.  The  actual  form  of  the  in- 
strument is  shown  in  Fig.  363. 

If  metallic  handles  that  can  be  grasped  with  the  hands  are  joined 
to  the  terminals  of  the  secondary  coil,  it  will  be  found  that  the 
shock  experienced  when  the  primary  circuit  is  made  is  much  lighter 
than  at  the  break.  When  the  primary  circuit  is  made,  the  current 
requires  a  large  fraction  of  a  second  to  attain  its  maximum  value  ; 
consequently  the  rate  of  change  of  the  lines  of  force  is  comparatively 
small.  At  the  break  of  the  primary,  however,  the  current  falls  to  zero 
very  suddenly,  thus  producing  a  high  rate  of  decrease  in  the  lines  of 
force.  For  this  reason  (§  446)  the  E.  M.  F.  is  correspondingly  greater 


ELECTRO-MAGNETIC   INDUCTION 


431 


at  the  interruption  of  the  primary  than  at  the  making  of  the 
circuit. 

The  induction  coil  is  widely  used  in  many  operations.  It  is  used 
in  gasoline  engines  to  ignite  the  mixture  of  air  and  vapor  (§271), 
for  igniting  explosives  in  blasting,  and  in  chemical  laboratories  for 
many  experimental  purposes.  Small  induction  coils  are  used  in  medi- 
cine for  the  remedial  effect  of  the  electric  current  and  in  the  speaking 
part  (transmitter  circuit)  of  the  modern  telephone.  Coils  of  large 
dimensions  are  employed  in  sending  messages  by  wireless  telegraphy 
and  in  the  production  of  X-rays. 

450.    An  Induced  E.M.F.  in  a  Moving  Conductor.  —  By 

referring  to  Fig.  360  it  becomes  clear  that  in  order  to 
change  the  number  of  lines  of  force  threading  through  the 
coil  of  wire,  the  conductor  has  to  cut  through,  or  across, 
some  of  these  lines.  The  following  experiment  will  show 
the  effect  of  this  operation  in  another  way. 

Hang  a  loose  copper  wire  across  the  classroom  and  connect  its  ends 
with  a  sensitive  galvanometer.  m  If  the  wire  be  set  swinging,  the  galva- 
nometer will  be  deflected  first  to  the  right  and  then  to  the  left  as  the 
conductor  cuts  through  the  earth's  lines  of  force.  Connect  one  of  the 
galvanometer  wires  to  the  center  of  the  swinging  conductor  and  repeat 
the  experiment.  The  induced  E.  M.  F.  will  be  about  one  half  as  great 
as  before,  as  shown  by  the  reduced  deflection  of  the  galvanometer. 

The  experiment  shows  that  when  a  conductor  cuts  across 
magnetic  lines  offeree,  an  E.M.F.  is  induced  within  it. 
If,  for  example,  the  con- 
ductor ab,  Fig.  364, 
move  from  right  to  left, 
and  the  galvanometer  be 
used  to  determine  the 
direction  of  the  E.M.  F., 
it  will  be  found  that  the 
E.  M.  F.  acts  through 
the  conductor  from  a 

towards    b:   i.e.    b   will 

.,      ...        .         FIG.  364.  — A  Conductor  ab  Cutting  Mag- 

act  temporarily  like  the  netic  Liues  of  Force.  \ 


432          A   HIGH   SCHOOL  COURSE   IN   PHYSICS 

positive  pole  of  a  cell,  and  a  like  the  negative  pole.  While 
the  wire  is  swinging  in  the  opposite  direction,  the  lines 
of  force  are  cut  from  the  opposite  side,  and  the  induced 
E.  M.  F.  is  reversed.  In  the  second  part  of  the  experi- 
ment the  lines  of  force  are  cut  only  one  half  as  fast,  and 
the  reduced  deflection  shows  that  the  value  of  the  E.  M.  F. 
varies  with  the  rate  at  which  the  lines  of  force  are  cut  by 
the  moving  conductor. 

It  is  clear  that  the  direction  of  the  induced  E.  M.F.  (or 
current)  depends  (1)  on  the  direction  of  the  lines  of  force 
and  (2)  on  the  direction  of  the  motion  of  the  conductor. 
The  three  quantities,  viz.  Motion,  Force,  and  Current,  have 
a  definite  relation  which  has  given  rise  to  the  following 
rule: 

Let  the  thumb  and  the  first  two  fingers  of  the  right  hand  be 
bent  at  right  angles  to  each  other,  as 
in  Fig.  365.  If,  now,  the  thumb 
point  in  the  direction  of  the  Motion 
of  the  wire,  and  the  First  finger 
point  in  the  direction  of  the  lines  of 
force  in  the  Field,  then  the  Center 
finger  indicates  the  direction  of  the 

FIG.    365.  —  Finding    the    Di-  p    *r _    f 
rection  of  an  Induced  Cur-  ^urreni;- 

The  key  to  the  rule  is  found  in 

the  corresponding  initial  letters  in  the  words  First  and 
Field,  Center  and  Current. 

2.   DYNAMO-ELECTRIC  MACHINERY 

451.  Principle  of  the  Dynamo.  —  The  laws  of  induced 
currents  find  their  most  important  application  in  the 
dynamo,  a  machine  which  facilitates  the  conversion  of 
mechanical  energy  into  electrical.  The  familiar  examples 
of  electric  street  lighting  and  the  operation  of  city  and 


JAMES    CLERK-MAXWELL    (1831-1879) 

Maxwell  ranks  as  one  of  the  greatest  of  mathematical  physicists 
on  account  of  the  important  practical  results  to  which  his  theories 
have  led.  He  was  born  in  Edinburgh,  Scotland,  where  he  received 
his  early  education.  In  1856  he  became  professor  of  physics  in 
Marischal  College  at  Aberdeen,  in  1860  professor  of  physics  and 
astronomy  in  King's  College  of  London,  and  in  1871  first  pro- 
fessor of  physics  in  Cambridge  University. 

Maxwell  built  upon  the  experimental  discoveries  of  Faraday,  so 
arranging  and  relating  them  as  to  make  them  yield  to  mathematical 
treatment.  He  advocated  the  view  that  electric  and  magnetic  forces 
result  from  certain  changes  in  the  distribution  of  energy  in  the 
ether.  He  showed  that  electro-magnetic  action  must  travel  through 
space  in  the  form  of  transverse  waves  and  with  the  velocity  of  light. 
Later  on  this  theory  was  corroborated  by  the  experiments  of  Hertz, 
who  was  first  to  produce  such  waves  and  show  that  they  could  be 
reflected,  refracted,  and  polarized  like  light.  By  these  two  physicists, 
the  one  attacking  problems  from  the  mathematical  standpoint,  the 
other  building  his  experiments  upon  the  theoretical  results  obtained, 
the  intimate  relation  between  light  and  electricity  has  been  amply 
confirmed. 

Among  other  subjects  investigated  by  Maxwell  was  the  kinetic 
theory  of  gases.  The  results  were  published  with  others  in  a 
memorial  edition  of  two  volumes  by  the  Cambridge  Press. 

Maxwell's  works  also  include  his  Theory  of  Heat  (1871),  Elec- 
tricity and  Magnetism  (1873),  and  a  clear  and  concise  treatise  of 
dynamics  entitled  Matter  and  Motion  (1876). 


OF  THE 

UNIVERSITY 

OF 


ELECTRO-MAGNETIC   INDUCTION 


» 

433 


FIG.  366.  —  Illustrating  the 
Dynamo  Principle. 


interurban  railroads  have  been  brought  about  by  the  devel- 
opment of  the  dynamo. 

Make  a  rectangular  coil  of  200  or  300  turns  of  very  small  copper 
wire  of  such  dimensions  as  to  rotate  between  the  poles  of  a  horse- 
shoe magnet.  Connect  the  ends  with  a 
d'Arsonval  galvanometer  and  place  the  coil 
in  the  magnetic  field  as  shown  in  Fig.  366. 
Starting  with  the  plane  of  the  coil  at  right 
angles  to  the  lines  of  force,  i.e.  vertical,  ro- 
tate it  through  90.°  A  deflection  will  show 
the  presence  of  an  induced  current.  Con- 
tinue the  rotation  through  the  next  90°.  A 
deflection  in  the  same  direction  will  be  ob- 
served. If,  now,  the  rotation  be  continued, 
a  deflection  in  the  opposite  direction  will 
be  produced  until  the  coil  has  returned  to 
its  initial  position. 

When  the  coil  is  revolved  from 
the  position  shown  in  Fig.  367,  the  two  portions  a  and  b 
begin  to  cut  the  lines  of  force  of  the  magnet  and  thus  de- 
crease the  number  of  lines  passing  through 
the  coil.  A  current  is  therefore  set  up 
by  the  induced  E.  M.  F.,  which,  accord- 
ing to  the  rule  given  in  the  preceding 
section,  is  upward  in  a  and  downward  in 
b.  For  the  first  180°  of  rotation,  the  por- 
tion a  of  the  coil  continues  to  cut  the 
FIG.  367.  — Loops  of  lines  of  force  in  the  same  direction  ;  and 

Wire    Perpendicu-  . 

lar   to   Lines   of  the  same  is  true  of  portion  6;  hence  the 

Force-  induced  current  thus  far  is  continuous 

and  in  one  direction.     When,  however,  the  coil  begins  to 

revolve  through  the  second  180°,  each  part  of  it  begins 

to  cut  magnetic  lines  in  the  opposite  direction;  hence  the 

induced  current  will  accordingly  flow  through  the  wire  in 

the  opposite  direction.     It  is  now  obvious  that  a  continuous 

rotation  of  the  coil  develops  a  current  whose  direction  reverses 

29 


434         A  HIGH   SCHOOL   COURSE   IN   PHYSICS 

every  time  the  plane  of  the  coil  is  perpendicular  to  the  lines 
.of  force.     Such  a  current  is  called  an  alternating  current. 
452.    The   Alternating-current  Dynamo. — Alternating- 
current  dynamos,  or  alternators,  are  based  upon  the  prin- 
x  ciples  of  induction  as  shown  by 

the  experiment  in  the  preceding 
section.  In  its  simplest  form, 
the  machine  consists  of  a  coil  of 
several  turns  of  wire  arranged 
to  rotate  between  the  poles  of  a 
strong  electro-magnet,  as  shown 
in  Fig.  368.  The  wire  is  wound 
FIG.  368. —Diagram  of  a  Simple  On  a  core  of  soft  iron  C  and 

Alternating-current  Dynamo.  ,  ,     ,  ,      ~      .         ~, 

mounted  on  a  shaft  A.  The  re- 
volving coil  is  called  the  armature,  and  the  electro-magnet 
whose  poles  are  N  and  S  is  called  the  field  magnet. 
In  order  to  provide  for  the  flow  of  the  induced  current  to' 
and  from  the  rotating  coil,  the  ends  of  the  wire  of  the  ar- 
mature are  connected  with  the  two  insulated  metal  collect- 
ing rings  a  and  a'.  Stationary  "  brushes,"  b,  5,  rest  against 
these  rings  as  shown  and  conduct  the  induced  current  to 
the  external  circuit  X.  The  efficiency  of  the  machine  is 
greatly  increased  by  the  presence  of  the  iron  core  0,  since 
it  multiplies  and  concentrates  the  number  of  magnetic 
lines  of  force  between  the  poles  -ZV  and  S. 

When  the  armature  is  rotated,  an  induced  alternating 
current  is  set  up  in  the  coil,  and  through  the  rings  and 
brushes  is  transmitted  to  the  external  circuit.  The  E. 
M.  F.  will  be  determined  by  the  number  of  lines  of  force 
cut  per  second.  This  will  of  course  be  greatest  when  the 
coils  are  moving  at  right  angles  to  the  magnetic  lines,  i.e. 
when  the  plane  of  the  coils  is  horizontal.  The  E.  M.  F. 
is  zero  at  the  instant  of  reversal,  for  at  that  time  all  por- 
tions of  the  coil  are  moving  parallel  to  the  lines  of  force. 


ELECTRO-MAGNETIC   INDUCTION 


435 


Hence,  the  E.  M.  F.  rises  from  zero  to  a  maximum  value 
and  then  decreases  to  zero  again,  at  which  time  it  reverses. 
Such  a  current  is  shown 
diagrammatically  by  the 
curve  in  Fig.  369. 


FIG.  369. — Diagram  of  an  Alternating 
Current. 


FIG.  370.  —  Diagram  of  a  Multipolar 
Machine. 


453.  Multipolar    Ma- 
chines. —  Alternators  are 

of  great  commercial  value  in  generating  electricity  for  both 
lighting  and  power.  They  have  largely  replaced  the  so- 
called  direct-current  dy- 
namos which  have  had  a 
world- wide  use.  But  for 
practical  purposes  it  is 
desirable  that  the  rever- 
sals of  an  alternating 
current  attain,  or  even 
exceed,  120  per  second. 
This,  of  course,  cannot  be 
accomplished  with  the 
two-pole  machine  described  in  the  preceding  section.  In 
the  multipolar  machine  there  are  several  poles  arranged 
around  the  armature  as  shown  in  Fig.  370.  In  this  ma- 
chine there  are  as  many  coils  of  wire  in  the  armature  as 
there  are  poles  in  the  field  magnets. 

As  the  moving  coils  cut  through  the  lines  of  force,  an 
induced  E.  M.  F.  is  set  up  in  each.  The  several  coils  are 
so  connected  that  the  E.  M.  F.  at  the  brushes  is  the  sum 
of  that  produced  in  the  separate  coils.  The  field  magnets 
are  excited  by  means  of  a  continuous  direct  current  from 
a  small  dynamo  (§  454).  Figure  371  shows  a  type  of 
multipolar  alternator  which  is  in  common  use. 

454.  The  Direct-current  Dynamo.  —  The  alternating-cur- 
rent dynamo  described  in  §  452  may  be  employed  to  pro- 
duce a  unidirectional  current  by  the  introduction  of  a 


436          A  HIGH   SCHOOL   COURSE   IN   PHYSICS 


so-called  commutator.     The  commutator  in  this  case  con- 
sists of  a  metal  ring  divided  into  two  semicircular  parts 


FIG.  371.  —  A  Multipolar  Alternator. 

a  and  a! ,  Fig.  372,  called  segments.  These  are  insulated 
from  each  other  and  mounted  on  the  shaft  which  car- 
ries the  armature.  Each  of  the  two  ends  of  the  arma- 
ture coil  connects  with  a  seg- 
ment of  the  commutator  as 
shown  in  the  figure.  The 
brushes  b  and  bf  which  conduct 
the  induced  current  to  and  from 
the  external  circuit  X  are  set  on 
opposite  sides  of  the  commuta- 
FIG.  372.— Diagram  of  a  Direct-  tor.  Their  position  is  very  im- 
portant. They  must  be  so  placed 

that  each  brush  changes  its  point  of  contact  from  one  segment 
to  the  other  at  the  instant  the  current  reverses  in  the  armature 
coil.  Thus  5,  for  example,  will  continually  be  in  contact 


ELECTRO-MAGNETIC  INDUCTION 


437 


(0 


FIG.  373.  —  (J)  Diagram  of  the  Armature 
Current.  (2)  Diagram  of  the  Current 
Taken  from  the  Brushes. 


with  a  positively  charged  segment,  and  br  with  a  negatively 
charged  one.  Hence,  in  this  case,  the  current  will  always 
flow  into  the  external 
circuit  through  b  and 
return  to  the  armature 
through  bf.  Such  a 
current  is  pulsating  in 
nature,  since  the 
E.  M.  F.  falls  to  zero 
twice  during  each  revo- 
lution (§  452).  A  comparison  of  the  alternating  current 
in  the  armature  with  the  pulsating  current  taken  off  at  the 
brushes  is  made  in  Fig.  373. 

455.  Dynamos  for  Steady  Currents.  —  The  pulsating  cur- 
rents produced  by  dynamos  of  one  coil  (Fig.  373)  are  un- 
satisfactory for  most  purposes.  The  difficulty  may  be 
overcome  and  a  continuous  current  developed  by  the  use 
of  several  coils  distributed  uniformly  over  the  armature. 

The  first  armature 
wound  in  this  manner 
was  the  so-called 
Gramme  ring,  invented 
by  a  Frenchman  in  1870. 
The  core  of  the  arma- 
ture is  an  iron  ring  so 
mounted  on  an  axle  as  to 
turn  between  the  poles, 
N  and  S,  Fig.  374,  of  a  strong  electro-magnet.  The  ring 
is  wound  with  several  coils  of  copper  wire  placed  at  equal 
distances.  These  are  represented  in  the  figure  by  the  sin- 
gle turns  numbered  from  1  to  12.  At  each  junction  of 
two  adjacent  coils  a  connection  is  made  with  a  segment  of 
the  commutator  (7,  which  consists  of  as  many  insulated 
bars  as  there  are  coils  in  the  armature.  The  lines  of  force 


From  the  Exte 


Circuit 

FIG.  374.  —  Diagram  of  the  Gramme 
Ring  Armature. 


438 


A   HIGH   SCHOOL   COURSE   IN    PHYSICS 


follow  through  the  iron  of  the  ring  from  N  to  $,  as  shown 
by  the  dotted  lines ;  hence  the  outer  portions  of  each  coil 
are  the  only  ones  that  cut  the  lines  when  the  armature  re- 
volves. 

If,  now,  the  armature  is  revolved  in  the  direction  shown 
by  the  arrow  at  the  top,  the  conductors  from  1  to  5  cut 
through  the  lines  in  a  downward  direction ;  hence  the 
E.M.  F.  throughout  these  coils  is  in  the  direction  shown 
by  the  arrowheads  on  the  wire  (§  450).  At  the  same 
time  the  conductors  from  7  to  11  are  cutting  the  lines  in 
an  upward  direction,  which  develops  an  E.M.F.  within 
them  as  shown  by  the  affixed  arrows.  An  inspection  of 
the  figure  will  show  that  upon  both  the  right  and  the  left 
side  of  the  armature  the  tendency  is  to  raise  the  lowest 
segment  of  the  commutator  to  a  high  potential  and  to 
reduce  the  topmost  one  to  a  low  potential.  Therefore, 
by  placing  the  brushes  b  and  br  in  contact  with  these 
points,  a  direct  current  is  led  through  the  external  cir- 
cuit in  the  direction 
shown  in  the  figure.  The 
E.  M.  F.  produced  by  such 
an  armature  is  practically 
constant  when  the  rate  of 
rotation  is  uniform. 

456.  The  Drum  Arma- 
ture. — The  modern  direct- 
current  dynamo  is  pro- 
vided with  an  armature 
ture  of  the  "drum"  type. 
This  consists  of  a  cylindrical  core,  or  drum,  of  iron 
upon  which  are  wound  numerous  coils  of  wire  equally 
spaced.  The  construction  is  made  clear  by  a  study  of 
Fig.  375.  If  the  windings  are  traced,  the  coils  will  be 
found  to  be  joined  in  series,  and  at  four  points  connec- 


FIQ.  375.  —  Diagram  of  the  Drum  Arma 
ture. 


ELECTRO-MAGNETIC  INDUCTION 


> 

439 


tions  are  made  with  the  segments  of  the  commutator  0. 
If  the  armature  is  now  rotated  between  the  poles  of  a 
magnet,  the  E.  M.  F.  will  at  no  time  be  zero  at  the 
brushes ;  for  at  every  instant  some  of  the  conductors  are 
cutting  magnetic  lines.  By  setting  the  brushes  at  the 
proper  points,  a  direct  and  fairly  steady  current  will  be 
transmitted  to  the  external  circuit. 

457.  Field  Magnets. — The  magnetic  poles  between 
which  the  armature  of  a  dynamo  rotates  receive  their  ex- 
citation from  coils  of  wire  carrying  an  electric  current. 
In  direct-current  machines  this  current  is  produced  by 
the  dynamo  itself.  The  initial  current  developed  in  the 
armature  depends  upon  the  residual  magnetism  (§  379) 
of  the  pole  pieces,  which  serves  to  induce  a  small  current. 
This  in  turn  increases  the  magnetism  until  the  poles 
finally  reach  their  full  strength. 

In  the  series-wound  'dynamo,  (1),  Fig.  376,  the  entire 
current  is  led  from  the  brushes  through  the  few  turns  of 


Main  Circuit 


Main  Circuit 


(0 


FIG.  376. —  (1)   Series-wound  Dynamo.     (2)    Shunt-wound  Dynamo. 
(3)   Compound-wound  Dynamo. 

thick  wire  of  the  field  magnets,  which  are  joined  in  series 
with  the  external  circuit  as  shown. 

In  the  shunt-wound  dynamo,  (2),  Fig.  376,  only  a  portion 
of  the  entire  current  is  led  through  the  many  turns  of  rather 


440 


A  HIGH   SCHOOL  COURSE   IN   PHYSICS 


fine  wire  in  the  coils  of  the  field  magnet,  while  the  main 
portion  is  conducted  through  the  external  circuit.  The 
field  magnet  coils  thus  form  a  shunt  (§  445)  to  the  ex- 
ternal circuit. 

In  the  compound-wound  dynamo,  (3),  Fig.  376,  the  field 
magnets  are  wound  with  two  coils,  one  being  joined  in 
series  with  the  external  circuit,  and  the  other  connected 
as  a  shunt  between  the  brushes.  A  compound-wound 
machine  adjusts  itself  to  variations  in  the  resistance  of 
the  external  circuit  in  such  a  way  as  to  maintain  a  con- 
stant difference  of  potential  between  the  brushes. 

458.  The  Acyclic,  or  Unipolar,  Generator. — It  is  of  in- 
terest to  note  that  the  alternations  of  the  current  induced 
in  the  armature  of  a  dynamo  can  be  prevented  by  employ- 
ing a  mechanism  of  the  proper  form.  Figure  377  shows  a 
sketch  of  a  so-called  acyclic,  or  unipolar,  generator,  which 


FIG.  377.  —  Diagram  of  an  Acyclic  Generator. 

consists  of  two  collecting  rings  R  and  R1  joined  together 
by  conducting  bars  1,  2,  3,  etc.  These  bars  are  attached 
at  their  centers  to  a  revolving  shaft  Z>,  but  are  insulated 
from  it.  Magnets  are  placed  with  their  poles  as  shown 
at  JVand  S.  When  the  armature  is  revolved  in  the  direc- 


ELECTRO-MAGNETIC  INDUCTION 


441 


tion  shown  by  the  arrows  MM[,  conductor  1  cuts  through 
the  field  of  the  magnets,  and  an  E.  M.  F.  is  induced  in  it 
(§  450)  in  the  direction  shown  by  the  arrow  at  (7.  Hence 
a  current  may  be  taken  from  the  brush  at  B ',  which 
rests  continually  against  the  collecting  ring,  through  the 
external  circuit  X  back  to  the  brush  B.  Since  each  con- 
ductor cuts  the  magnetic  lines  of  force  in  the  same  direc- 
tion, B'  is  always  the  positive  brush  and  B  the  negative, 
and  hence  a  direct  current  flows  through  both  the  inter- 
nal and  external  parts  of  the  circuit. 

In  the  commercial  form  of  this  type,  Fig.  378,  the  mag- 
netic field  extends  completely  around  the  armature  and  is 


FIG.  378.  —  An  Acyclic  Generator. 

excited  by  means  of  a  current  in  the  field  coils  FF  which 
lie  in  concentric  circles  around  the  cylindrical  steel  core 
A  A.  The  armature  conductors  1,  #,  #,  etc.,  are  attached 
to  the  periphery  of  a  steel  cylinder  from  which  they  are 
insulated. 

Unipolar  generators  are  designed  to  run  at  a  high  speed, 
and  a  desired  E.M.F.  is  obtained  by  arranging  several 


442         A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

independent  sets  of  conductors  and  their  corresponding 
collecting  rings  in  the  armature  of  the  machine  and  join- 
ing their  brushes  in  series.  In  this  manner  the  E.  M.  F.s 
of  the  separate  sets  are  added  together.  The  greatest 
advantages  of  this  form  of  generator  over  that  of  the 
commutator  type  are  the  elimination  of  commutator  diffi- 
culties and  lower  cost  of  construction. 

EXERCISES 

1.  In  order  to  show  that  a  current  of  electricity  is  produced  when 
a  magnet  is  thrust  into  a  coil  of  wire,  why  is  it  usually  necessary  to 
employ  a  coil  of  many  turns? 

2.  Account  for  the  enormous  difference  of  potential  induced  be- 
tween the  terminals  of  the  secondary  of  an  induction  coil? 

3.  Connect  one  terminal  of  a  spark  coil  (induction  coil)  to  the 
outer  coating  of  a  Leyden  jar  and  bring  the  other  close  to  the  knob. 
Put   the  coil  in  operation  and  show  that  the  jar  becomes  charged. 
Explain  how  the  charge  can  "jump"  into  the  jar,  but  cannot  escape. 

4.  How  many  revolutions  per  minute  would  have  to  be  made  by 
a  two-pole  alternator  to  produce  120  alternations  of  the  current  per 
second  ? 

5.  An  alternator  has  16  poles.     How  many  alternations  per  second 
will  be  produced  when  the  speed  is  375  revolutions  per  minute? 

6.  What  will  be  the  effect  upon  the  E.  M.  F.  of  a  shunt-wound 
dynamo  of  introducing  resistance  into  the  field  magnet  circuit? 

SUGGESTION.  —  Consider  the  effect  of  the  added  resistance  on  the 
current  in  the  field  coils  and  also  on  the  magnetic  field. 

7.  Does  this  suggest  a  way  in  which  the  E.  M.  F.  of  a  shunt-wound 
machine  can  be  regulated  ? 


3.  TRANSFORMATION  OF  POWER  AND  ITS  APPLICATIONS 

459.  The  Electric  Motor.  —  Electrical  energy  is  trans- 
formed into  mechanical  by  means  of  electric  motors.  The 
direct-current  motor  does  not  differ  greatly  in  construction 
from  the  dynamo.  The  principle  underlying  its  action  is 
illustrated  by  the  following  experiment. 


ELECTRO-MAGNETIC   INDUCTION 


443 


Suspend  a  wire  on  the  apparatus  described  in  §  412  so  that  its 
lower   end   just  dips   into  mercury.     Hold   a  horseshoe  magnet  as 
shown  in  Fig.  379  and  send  a      /-tfftrx^ 
current  from  a  new   dry  cell 
through    the  wire.     The   wire 
will  be  found  to  move  at  right 
angles  to  the  lines  of  force,  as 
shown  by  the  arrow.    If  the  cur- 
rent is  nowre  versed,  the  wire  will 
move  in  the  opposite  direction. 


The  experiment  shows 
clearly  that  a  conductor  in 
which  a  current  is  flowing 
tends  to  move  in  a  direction 
at  right  angles  to  the  lines 
of  force  of  a  magnetic  field  FIG.  379.  — Movement  of  a  Conductor  in 

in  which  it  is  placed.     The  a  M*s*eiic  Field- 

motion  may  be  determined  by  applying  the  rule  of  §  450, 
but  by  using  the  left  hand  instead  of  the  right. 

Let  this  principle  be  applied  to  Fig.  374.  If  a  current- 
from  some  source  be  sent  through  the  armature  from  bf  to  b, 
the  current  through  the  coils  will  take  the  direction  indi- 
cated by  the  arrows,  dividing  as  it  leaves  br.  According  to 
the  principle  shown  in  the  experiment,  all  the  conductors 
from  1  to  5  will  be  urged  in  an  upward  direction,  and  those 
from  7  to  11,  downward.  Since  the  current  employed  by 
the  motor  flows  through  the  field  magnet  coils,  a  powerful 
magnetic  field  is  produced  in  which  the  armature  will  ro- 
tate with  sufficient  power  to  turn  the  wheels  of  factories 
and  propel  electric  cars,  launches,  automobiles,  etc. 

460.  The  Electric  Railway  Car.  —  A  familiar  applica- 
tion of  the  electric  motor  for  the  generation  of  mechanical 
power  is  found  in  the  electric  railway,  Fig.  380.  A 
current  from  the  generator  at  the  power  house  is  trans- 
mitted through  the  trolley  wire,  or  in  some  cases  through 


444          A   HIGH   SCHOOL   COURSE   IN   PHYSICS 

a  third  rail,  and  by  means  of  a  metallic  arm  A  to  the 
motors  placed  under  the  floor  of  the  car  at  M.  The  axles 
of  the  car  are  so  geared  to  those  of  the  motors  that  the 
car  is  propelled  by  the  rotation  of  the  motor  armatures. 

I  Feed  Wire  \  \       •< 


Trolley  Wire 


JJUUUUL 


•f 


Dynamo  or 


Track         — 


It 

FIG.  380.  —  Diagram  of  an  Electric  Railway  System. 

From  the  motors  the  current  returns  to  the  power  house 
through  the  track,  the  rails  being  carefully  "  bonded " 
together  with  copper  conductors.  In  circuit  with  the 
motors  is  placed  the  controller  (7,  operated  by  the  motor- 
man.  By  means  of  a  series  of  resistance  coils  connected 
with  the  controller,  the  current  can  be  increased  or  di- 
minished and  the  speed  of  the  car  thus  regulated.  An- 
other device  enables  the  motorman  to  reverse  the  motors. 
A  third  accessory  serves  in  applying  the  air  brakes 
(§  159)  to  check  the  speed  after  the  current  has  been 
completely  interrupted  at  the  controller. 

461.  The  Alternating-current  Transformer.  —  Alternat- 
ing currents  owe  their  extensive  application  to  the  fact 
that  the  potential  difference  between  two  points  can  be 
easily  reduced  from  a  dangerous  one  of  many  thousand 
volts  to  one  of  a  safe  value  for  dwelling-house  use  and 
for  many  other  purposes.  This  is  accomplished  by  means 
of  a  transformer,  which  is  simply  a  modified  induction 
coil. 

The  principle  involved  in  the  transformer  is  easily 
understood  from  a  study  of  Fig.  381.  An  iron  core  R 
is  wound  with  two  independent  coils  of  wire  P  and  8. 


ELECTRO-MAGNETIC   INDUCTION 


445 


Let  an  alternating  current  be  sent  through  the  primary 
coil  P,  and  let  the  secondary  be  connected  with  a  group 
of  lamps  L.  The  cur- 
rent in  P  magnetizes 
the  iron  core  in  one 
direction,  then  demag- 
netizes and  remagnet-  FIG.  381.— Diagram  of  a  Transformer. 

izes    it     again     in    the 

opposite  direction,  while  the  magnetic  lines  follow  the  iron 
core  through  the  secondary  coil  8.  As  a  result  of  these 
magnetic  changes  in  the  core  an  alternating  current  is  in- 
duced in  S  and  flows  through  the  lamps. 

If  there  are  more  turns  of  wire  on  the  secondary  coil 
S  than  on  the  primary  coil  P,  the  potential  difference  at 

the  terminals  of  S  will  be 
greater  than  at  P,  because 
of  the  larger  number  of 
loops  of  wire  in  which  the 
magnetic  changes  occur. 
In  this  case  the  transformer 
is  called  a  step-up  trans- 
former. If,  however,  the 
secondary  contains  the 
fewer  turns,  its  potential 
difference  is  lower  than  that 
of  the  primary,  and  the  transformer  is  a  step-down  trans- 
former. The  ratio  of  the  two  potential  differences  is 
equal  to  the  ratio  of  the  number  of  turns  of  wire  in  the  two 
coils.  For  example,  a  transformer  that  is  to  be  used  to 
reduce  a  voltage  of  2000  to  a  lower  one  of  100  volts  would 
be  constructed  by  winding  the  primary  coil  with  20  times 
as  many  turns  as  the  secondary.  The  same  transformer 
would  reduce  a  potential  difference  of  2200  volts  to  one 
of  110  volts. 


fHmary 


'Secondary 


FIG.  382.  — A  Commercial  Trans- 
former. 


446          A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

462.  Utility  of  the  Transformer.  —  The  value  of  the 
transforming  process  is  made  clear  by  the  study  of  a 
specific  case.  For  example,  electric  power  is  to  be  trans- 
ferred from  a  power  station  to  a  large  city  over  several 
miles  of  wire.  It  is  desired  that  the  number  of  amperes 
(  0  in  Eq.  8,  §  437)  be  small  in  order  to  reduce  to  a  mini- 
mum the  loss  of  energy  due  to  the  heating  of  the  con- 
ducting wires.  The  alternator  used  at  the  power  house 
develops  a  potential  difference  of  2000  volts.  This  cur- 
rent is  led  through  the  primary  coil  of  a  "  step-up " 

c 


FIG.  383.  —  The  Transformer  System  of  a  Long  Distance  Power  Circuit. 

transformer  A,  Fig.  383,  where  the  potential  difference 
is  raised  to  11,000  volts,  while  the  number  of  amperes 
becomes  proportionately  reduced.  At  this  voltage  the 
current  would  be  about  7  amperes  per  100  horse  power. 
Since  wires  of  so  great  a  potential  difference  are  unsafe 
to  lead  into  houses,  the  voltage  is  reduced  from  11,000  to 
2000  volts,  by  a  "step-down"  transformer  B  where  the 
line  enters  the  city,  and  again  at  the  houses,  from  2000  to 
100  volts  by  the  transformer  O. 

Since  the  power  transmitted  by^an  electric  current  is  the  product 
of  the  number  of  volts  and  amperes  (Eq.  7,  §  436),  a  current  of  10 
amperes  under  a  potential  difference  of  11,000  volts,  for  example, 
delivers  a  power  of  110,000  watts,  which  is  about  147.5  horse  power. 
It  is  plain,  therefore,  that  a  large  amount  of  power  under  a  high 
voltage  can  be  transferred  by  a  small  current.  The  power  generated 
at  Niagara  Falls  is  distributed  to  Buffalo,  Syracuse,  and  other  places 
with  potential  differences  of  from  22,000  to  60,000  volts. 

Again,  if  the  transformer  C,  Fig.  383,  deliver  a  current  of  5  amperes 
under  a  potential  difference  of  100  volts  for  lighting  a  building,  the 


ELECTRO-MAGNETIC  INDUCTION  447 

power  delivered  is  5  x  100,  or  500  watts.  In  the  long  distance  circuit 
between  A  and  B  where  the  voltage  is  11,000,  the  current  would  be 
500  -4-  11,000,  or  0.045  ampere.  Hence,  to  take  5  amperes  at  100  volts 
would  increase  the  current  in  the  long  line  only  0.045  ampere. 

The  heat  losses  in  long  distance  power  transmission  render  it 
impracticable  to  convey  large  currents,  as  shown  in  §  438.  But  the 
examples  above  show  that  by  raising  the  potential  difference  to  sev- 
eral thousand  volts  the  current  is  proportionately  reduced;  conse- 
quently a  given  power  can  be  transferred  with  far  less  heat  loss, 
since  the  heat  generated  is  proportional  to  the  square  of  the  current. 
It  is  in  this  manner  that  the  transformer  has  solved  the  problem  of 
long  distance  transmission  of  power  from  places  where  power  is  com- 
paratively cheap  to  distant  manufacturing  centers  where  it  would  be 
expensive.  The  distribution  of  power  over  long  electric  railway  lines 
is  accomplished  by  means  of  so-called  high  tension,  i.e.  high  potential, 
apparatus. 

The  alternating  current  transformer  affords  one  of  the 
most  striking  examples  of  the  transformation  and  trans- 
ference of  energy  to  be  found.  The  power  in  one  circuit 
is  transferred  to  another  entirely  without  any  mechanical 
connection  between  the  two.  It  is  necessary  only  that  the 
magnetic  lines  set  up  by  the  current  in  the  primary  coil 
pass  through  the  secondary.  No  mechanical  motion  is 
concerned  in  the  process.  The  power  loss  in  a  good 
transformer  is  usually  not  more  than  3  or  4  per  cent. 

463.  Incandescent  Lighting.  —  It  is  a 
familiar  fact  that  the  heating  effect  of  an 
electric  current  is  employed  in  the  process 
of  electric  lighting.  The  simplest  case 
to  study  is  the  incandescent  lamp.  See 
Fig.  384.  In  this  lamp  the  current  is 
sent  through  a  carbon  filament  (7,  which 
is  heated  to  incandescence.  In  order  to 
prevent  the  carbon  from  burning,  as  well  FIG.  384.  —  An  incan- 
as  to  prevent  loss  of  heat  by  convection,  Jj£JJ| 
it  is  inclosed  in  a  highly  exhausted  glass  and  Socket. 


448          A  HIGH   SCHOOL  COURSE  IN   PHYSICS 

bulb.  Connections  are  made  with  the  ends  of  the  filament 
by  means  of  two  short  pieces  of  platinum  wire  sealed  in 
the  glass.  One  of  these  leads  to  the  contact  A  in  the 
center  of  the  base,  the  other  to  the  brass  rim  B  which  holds 
the  lamp  in  its  socket  S.  Through  these  the  current  is 
transmitted  to  and  from  the  lamp. 

On  account  of  its  large  consumption  of  power  (about 
3.5  watts  per  candle  power),  many  efforts  have  been  made 
to  produce  lamps  of  higher  efficiency.  At  the  present 
time  the  carbon  filament  lamp  is  being  rapidly  replaced 
by  those  provided  with  metallic  filaments  of  the  rather 
uncommon  metals  tantalum  or  tungsten.  These  metals 
admit  of  being  drawn  out  into  very  thin  wires  which  can 
be  heated  white-hot  without  melting.  These  thin  fila- 
ments are  mounted  in  glass  bulbs  in  much  the  same  man- 
ner'as  the  carbon  filaments.  Not  only  is  the  light  which 
metallic  filament  lamps  emit  whiter  than  that  given  off  by 
the  carbon  filaments,  but  the  efficiency  is  far  greater.  In 
the  tungsten  lamp  the  consumption  is  about  1.25  watts  per 
candle  power. 

Ordinarily  the  potential  difference  on  a  lamp  circuit  is 
maintained  at  110  or  220  volts.  Lamps  are  adapted  to  the 
voltage  of  the  circuit  on  which  they  are  to  be  used.  A 
16-candle-power  lamp  with  a  carbon  filament  requires  a 
current  of  slightly  more  than  0.5  ampere  when  the  voltage 
is  110  and  about  0.25  ampere  when  the  voltage  is  220. 
The  power  necessary  is,  by  equation  (?),  §  436,  110  x  0.5, 
or  55  watts. 

The  Nernst  lamp  employs  a  short  rod,  or  "  glower,"  Fig.  385,  com- 
posed of  oxides  which  are  maintained  at  a  high  temperature  by  the 
passage  of  a  current  of  electricity.  Although  the  glower  is  a  non- 
conductor at  ordinary  temperatures,  it  becomes  a  conductor  when 
heated.  The  glower  is  mounted  close  to  a  heater  coil  of  fine  platinum 
wire  through  which  a  current  passes  when  the  lamp  is  turned  on. 


ELECTRO-MAGNETIC  INDUCTION  449 

As  soon  as  the  glower  becomes  sufficiently  hot,  the  heater  coils  are 
automatically  thrown  out  of  circuit,  and  the  current  then  flows  only 
through  the  glower.  Since  the  glower  is  in- 
combustible, it  can  be  used  without  being- 
inclosed.  The  efficiency  of  the  Nernst  lamp 
is  considerably  below  2  watts  per  candle 
power. 

464.  Incandescent  Lamp  Circuits.  — 

Incandescent  lamps  are  connected  in 
parallel  between  the  two  main  wires    FIG.  385.  —  Showing  Parts 
leading  into  a  building.     These  wires         of  a  Nernst  Lamp- 
are  maintained  by  the  dynamo  D,  Fig.  386,  at  a  potential 
difference  of  110  volts,  so  that  any  lamp  may  be  turned 

— 1 — | — - — - — «^       on  or  off  without  interfer- 

p       O  O  O  O  Df°y>    *ng  with  the  others.     Each 
— I 1 — I — I — I — — ^      16-candle-power  lamp   re- 
quires a  current  of  half  an 
ampere  and,  consequently, 
has  a  resistance  when  hot 

FIG.  386.  — Showing  the  Connection  of      of  220   ohms.      Lamps   are 
Incandescent  Lamps  in  a  Circuit.  often   joined  in  groups,  as 

shown  at  A.     In  this  case  the  switch  placed  at  S  controls 
all  the  lamps  of  the  group. 

465.  Cost  of  Electric  Power.  — Electrical  energy  is  sold 
at  a  certain  rate  per  watt-hour.     A  watt-hour  is  a  volt- 
ampere-hour  ;  in  other  words,  an  ampere  of  current  flow- 
ing under  a  potential  difference  of  one  volt  for  one  hour 
delivers  a  watt-hour  of  energy.     Hence  a  110-volt  lamp 
carrying  0.5  ampere  requires  110  x  0.5  x  1,  or  55  watt- 
hours  for  every  hour  it  is  used.     In  commercial  lighting 
a  meter  which  is  designed  to  register  the  consumption  of 
energy  in    kilowatt- hours  is  placed    in  the  circuit  at  the 
point  where  the  wires  enter  each  consumer's  house.     Thus 
readings  of  the  meter  show  the  amount  of  electrical  energy 

for  which' the  user  is  charged. 
30 


450          A  HIGH   SCHOOL   COURSE   IN   PHYSICS 

466.  The  Electric  Arc.  —  The  first  electric  arc  was  ex 
hibited  in  1809  by  the  great  English  scientist,  Sir  Hum- 
phry Davy.     For  this  purpose  Davy  employed  over  2000 
voltaic  cells  joined  to  two  pieces  of  charcoal  which  were 
touched  and  then  slightly  separated.    The  same  experiment 
may  be  easily  made  on  any  commercial  lighting  circuit. 

Wind  bare  copper  wire,  about  No.  18,  around  pieces  of  electric 
light  carbons.  Join  these  in  series  with  a  resistance  of  about  10  ohms 
of  iron  wire  to  the  terminals  of  a  110-volt  lighting  circuit.  The  re- 
sistance may  be  made  by  winding  150  to  200  feet  of  No.  18  or  19  iron 
wire  on  a  suitable  frame.  Touch  the  tips  of  the  carbons  together  and 
at  once  separate  them  about  a  quarter  of  an  inch.  An  intensely 
bright  light  will  be  produced  as  the  current  continues  to  flow  across 
the  gap. 

By  the  separation  of  the  carbon  rods,  a  high  tempera- 
ture is  produced  by  the  current,  which  vaporizes  some  of 
the  carbon,  forming  a  conducting 
layer  from  one  to  the  other.  The 
resistance  of  this  mass  of  vapor  may 
not  be  more  than  3  or  4  ohms.  If 
the  current  is  a  direct  one,  the  tem- 
perature of  the  positive  carbon  rises 
above  that  of  the  negative,  and  from 
it  comes  the  greater  portion  of  the 
light.  In  this  case  the  positive  car- 
bon is  consumed  about  twice  as  fast 
as  the  negative  and  becomes  hollowed 
out,  as  in  Fig.  387,  while  the  negative 
remains  pointed.  When  an  arc  is 
produced  by  an  alternating  current, 
FIG.  387.  — The  Electric  light  is  given  out  equally  from  the 

Arc  Produced  by  a  Di-  J 

rect  Current.  two  points,  and  the  two  rods  are  con- 

.sumed  at  the  same  rate. 

467.  Arc  Lamps.  —  With  the  development  of  the  dy- 
namo, the  arc  lamp  as  a  means  of  illumination  has  come 


ELECTRO-MAGNETIC  INDUCTION 


451 


FIG.  388.  — The  Hand-feed  Electric 
Lamp. 


into   extensive   use.      In   the   so-called   hand-feed   lamp, 

Fig.  388,  the  positions  of  the  carbons  are  controlled  by 

the  screw  heads  S.     They  are 

at  first   permitted    to   touch 

and  then  are  separated.  As 
fast  as  the 
rods  are 
consumed, 
they  are 
slowly  fed 
together 
by  the  op- 
erator. Such  lamps  are  used  mainly  in 
projection  lanterns  (§  324).  In  the  arc 
lamp  of  automatic  feed,  Fig.  389,  the 
mechanism  has  two  duties  to  perform: 
(1)  to  separate  the  carbons  when  the 
current  starts,  and  (2)  to 
feed  the  carbons  together 
as  fast  as  they  are  con- 
sumed, always  keeping  the 

FIG.  389.  -The  Auto-     PrOPflr    8PaC6     between 
matic  Arc  Lamp.          them. 

The  consumption  of  the  carbon  rods  in  the  arc  lamp 
can  be  largely  reduced  by  inclosing  the  arc  in  a  globe 
that  is  nearly  air-tight  as  shown  in  Fig.  390.  In  this 
form  of  lamp  the  carbon  burns  from  60  to  100  hours. 

In  the  lamps  just  described  the  light  is  emitted  by 
the  incandescent  ends  of  the  carbons.  If  the  carbons,  -c, 

r  IG.     o9U.  —  111- 

however,  are  cored   with   a   mixture   of   carbon   and          closed  Elec- 
metallic  salts,  a  highly  luminous  vapor  is  maintained         trie       Arc 
by  the  current  between  the  two  terminals.     In  flaming         Lamp. 
arc  lamps  the  carbons  are  cored  with  calcium  salts  which  serve  to  give 
the  light  a  bright  yellow  color. 

The  open  arc  is  operated  by  a  current  of  from  5  to  10  amperes  and 
45  to  50  volts,  and  the  candle  power  in  the  direction  of  greatest  in- 


452          A  HIGH  SCHOOL  COURSE  IN   PHYSICS 

tensity  is  about  1000.     The  inclosed  arc  lamp  requires  a  voltage  of 
about  80  and  a  current  of  from  5  to  8  amperes. 

EXERCISES 

1.  According  to  Lenz's  law,  in  what  direction  in  a  circuit  will  a 
current  be  induced  by  the  sudden  interruption  of  a  current  in  a  neigh- 
boring conductor  ? 

2.  If  the  experiment  of  §  450  be  repeated  with  a  long  loop  of  wire, 
it  will  be  found  that  no  deflection  of  the  galvanometer  will  be  produced 
by  swinging  both  parts  of  the  loops  together  across  the  earth's  lines  of 
force.     Explain.     By  swinging  the  two  parts  of  the  loop  in  opposite 
directions,  a  large  deflection  is  obtained.     Why? 

3.  Would  you  class  the  induction  coil  as  a  "  step-up  "  or  a  "  step- 
down"  transformer? 

4.  Explain  why  the  interrupter  is  an  accessory  part  of  the  induc- 
tion coil  but  not  of  the  transformer. 

5.  A  transformer  carries  a  current  of  10  amperes  in  its  primary 
coil,  under  a  potential  difference  at  its  terminals  of  2000  volts.     If  it 
delivers  a  current  of  194  amperes  with  a  potential  difference  of  100 
volts,  how  much  energy  is  transformed  per  hour,  how  much  delivered, 
and  how  much  wasted? 

6.  If  a  building  contained  50  110-volt  incandescent  lamps,  what 
voltage  would  have  to  be  supplied  to  operate  them  if  they  were  all 
joined  in  series?     Show  that  this  would  be  impracticable. 

7.  Show  by  a  diagram  the  manner  of  connecting  ten  16-candle- 
power  110-volt  incandescent  lamps  in  parallel.     What  would  have  to 
be  the  voltage  and  how  much  current  would  be  required  ? 

SUGGESTION.  —  Consider  that  in  the  case  of  parallel  conductors  the 
total^urrent  is  the  sum  of  that  in  the  separate  parts. 

8.  Why  would  the  alternating-current  transformer  be  entirely  in- 
effective in  transforming  a  continuous  current? 

9.  Compute  the  monthly  cost  of  operating  five  1 6-candle-power  in- 
candescent lamps  when  current  costs  10  ct.  per  kilowatt-hour,  allow- 
ing an  average  use  of  3  hr.  per  day  and  3£  watts  per  candle  power. 

10.  Compute  the  monthly  cost  of  current  for  an  open  arc  lamp  re- 
quiring 8  amperes  at  50  volts,  allowing  3  hr.  per  day  and  10  ct.  per 
kilowatt-hour. 

11.  If  the  potential  difference  between  the  trolley  of  an  electric 
railway  and  the  track  is  550  volts,  show  how  it  would  be  possible  to 


ELECTRO-MAGNETIC   INDUCTION 


453 


operate    110- volt  incandescent  lamps  by  properly  connecting  them 
together.     Diagram  the  system. 

12.  Show  how  a  workman  standing  on  the  top  of  an  electric  car 
can  safely  handle  the  trolley  wire  with  bare  hands,  while  one  standing 
on  the  ground  would  be  severely  shocked  by  coining  in  contact  with 
any  wire  forming  a  connection  with  the  trolley  wire. 

13.  25  street  lamps  each  operated  by  a  potential  difference  at  its 
terminals  of  45  volts  are  joined  in  series.     What  potential  difference 
must  be  maintained  at  the  terminals  of  the  dynamo  ?     What  would 
have  to  be  the  current  strength?     Disregard  the  line  loss. 

14.  What  has  been  the  effect  of  economical  long  distance  powei 
transmission  upon  manufacturing  industries,  railway  development, 
etc.? 

4.  THE  TELEGRAPH  AND  THE  TELEPHONE 

468.  The  Morse  Telegraph.  —  The  most  extensive  use 
of  the  magnetic  effect  of  electric  currents  is  made  in  the 
telegraph  systems  in  common  use.  In  1831  Joseph  Henry 
produced  audible  signals  at  a  distance,  but  the  system 
generally  employed  in  this  country  is  that  designed  by 
Samuel  F.  B.  Morse  and  first  used  in  1844.  The  instru- 
ments found  at  each  station  are  the  key,  sounder,  and  re- 
lay. 

The  key,  Fig.  391,  is 
merely  a  convenient 
device  for  making  and 
breaking  an  electric  cir- 
cuit at  A  by  operating 
the  lever  L.  It  is  also 
provided  with  a  switch 
S,  so  that  the  circuit 
may  be  left  closed  when 
the  key  is  not  in  use. 

The  sounder,  Fig. 
392,  consists  of  an  elec- 
tro-magnet M  which  at-  FIG.  392.  — Telegraph  Sounder. 


FIG.  391.  —  Telegraph  Key. 


454          A   HIGH   SCHOOL   COURSE  IN   PHYSICS 

tracts  the  iron  armature  A  whenever  a  current  is  sent 
through  it,  thus  causing  the  heavy  brass  bar  B  to  be 
drawn  down  against  (7  with  a  sharp  click.  When  the  cur- 
rent is  interrupted,  the  armature  is  no  longer  attracted, 
and  the  bar  is  lifted  against  D  by  an  adjustable  spring. 
Transmitted  messages  are  read  by  ear  from  the  clicks  of 
this  instrument. 

469.  Plan  of  a  Short  Telegraph  Line.  —  Short  lines  in 
which  the  resistance  is  small  require  at  each  station  only 
the  key  and  sounder,  as  in  Fig.  393.  The  connection  is 

Sounder  Sounder  Key , 

Line  Battery          TV/H wY* 

|.|i|i|i|i|i HJL     ~~L^  r-*-1         Slult°h 


Station  A  Maln  Lln0  Station  B 


Earth  •== 

Earth 

Fro.  393.  —  A  Short  Telegraph  Line. 

usually  made  by  a  single  wire,  the  circuit  being  completed 
by  joining  a  wire  at  each  end  to  a  metal  pipe  or  to  a  metal 
plate  buried  in  the  ground.  Thus  the  earth  forms  a  part 
of  the  circuit.  The  battery  may  be  placed  anywhere  in 
the  line. 

When  the  operator  at  Station  A  wishes  to  send  a  mes- 
sage to  Station  B,  he  opens  the  switch  on  his  key,  which 
breaks  the  circuit.  The  sounder  cores  are  thus  demag- 
netized, and  the  bars  are  thrown  up  by  the  springs.  Now, 
by  means  of  the  key,  the  operator  A  can  make  and  break 
the  circuit  which  will  cause  both  sounders  to  click  off  the 
"  dots  "  and  "  dashes  "  composing  the  message.  A  dot  is 
produced  by  a  quick  stroke  of  the  key  which  closes  the 
circuit  for  only  an  instant ;  a  dash  is  a  slower  stroke 
which  leaves  the  circuit  closed  for  a  slightly  longer  time. 


ELECTRO-MAGNETIC  INDUCTION  455 

The  operator  at  B,  skilled  in  the  interpretation  of  the 
clicks  of  the  sounder,  reads  and  records  the  message. 
Since  a  telegraph  circuit  is  left  closed  when  not  in  use,  a 
"  closed  circuit,"  cell  like  the  Daniell  or  gravity  must  be 
used. 

The  Morse  code  given  below  is  composed  entirely  of 
dots,  dashes,  and  spaces.  A  small  space  is  left  between 
letters  and  a  slightly  longer  one  between  words. 

THE  MORSE  TELEGRAPH  CODE 

a  -  —  h o--  u 

b i--  p v 

c---  j .  <1  •  •  —  •  w 

d k r x 

e  -  1  •  s y 

f m t  —  z  —  - 

g U__- 

470.  The  Relay  and  Its  Use.  —  In  long  telegraph  lines 
on  which  there  are  many  instruments,  the  resistance  is 
usually  so  great  that 
the  current  in  the  main 
line  is  too  feeble  to  op- 
erate the  sounders  with 
sufficient  loudness.  The 
difficulty  is  avoided  by 
the  use  Of  the  relay,  FIG.  S94.- The  Telegraph  Relay. 

which  is  a  more  sensitive  instrument  than  the  sounder. 
The  relay,  Fig.  394,  consists  of  an  electro-magnet  M  con- 
taining several  thousand  turns  of  wire  (about  150  ohms) 
which  is  placed  in  the  main  line  at  each  station  together 
with  the  key.  The  armature  and  bar  H  of  the  relay  are 
made  very  light,  and  all  the  adjustments  of  the  instru- 
ment may  be  made  with  great  precision.  The  function 
of  the  relay  is  to  open  and  close  at  K  a  local  circuit  which 
contains  merely  the  sounder  and  two  or  three  cells.  It 


456 


A  HIGH   SCHOOL   COURSE  IN   PHYSICS 


will  readily  be  seen  that  since  there  is  little  resistance  to 
reduce  the  current,  very  distinct  clicks  will  be  produced 
on  the  sounder  by  the  current  from  this  local  battery  every 
time  the  relay  automatically  closes  and  opens  the  local 
circuit. 

471.  The  Long  Distance  Telegraph  System.  — The  usual 
arrangement  of  the  parts  of  a  telegraph  system  is  shown 
diagrammatically  in  Fig.  395.  In  sending  a  message  irom 


FIG.  395.  —  A  Long  Distance  Telegraph  System. 

Detroit  to  Buffalo,  for  example,  the  Detroit  operator  uses 
precisely  the  method  described  in  §  469.  When  the  circuit 
is  opened  at  Detroit,  no  current  flows  from  the  main  line 
battery,  and  all  the  relays  on  the  line  release  their  arma- 
tures and  thus  open  every  local  circuit.  When  the  De- 
troit operator  presses  his  key,  the  main  line  battery  sends 
a  current  out  over  the  line,  and  the  electro-magnets  of  the 
relays  draw  the  armatures  down  and  thus  close  the  local 
circuits,  causing  every  sounder  to  produce  a  sharp  click. 
Thus  the  dots  and  dashes  comprising  a  message  may  be 
read  by  every  operator  along  the  line. 

The  relay  may  also  be  used  to  repeat  a   message   to 


ELECTRO-MAGNETIC   INDUCTION 


457 


another  line  instead  of  transmitting  it  to  a  short  local 
circuit.  A  message  from  Chicago  to  New  York  may  be 
repeated  to  a  Detroit-Buffalo  line  at  Detroit,  to  a  Buffalo- 
Syracuse  line  at  Buffalo,  and  finally  to  a  Syracuse-New 
York  line  at  Syracuse.  Since  the  repeating  is  performed 
automatically  at  each  of  these  stations,  no  time  is  lost  in 
the  transfer  from  one  line  to  another.  The , relay  when 
used  in  this  manner  is  called  a  repeater. 

472.  The  Telephone.  —  The  simplest  manner  in  which 
speech  produced  at  one  station  can  be  reproduced  by  elec- 
trical means  at  another  is  by  means  of  two  telephone 


Line 


FIG.  396.  —  A  Simple  Telephone  System. 

"  receivers  "  connected  by  two  wires,  or  by  one  wire  and 
the  earth.  See  Fig.  396.  The  telephone  receiver  was 
invented  in  1876  almost  simultaneously  by  Alexander 
Graham  Bell  and  Elisha 
Gray,  both  Americans.  It 
consists  simply  of  a  perma- 
nent bar  magnet  M,  Fig.  397, 
surrounded  at  the  end  by  a 

Coil   of  fine   insulated   copper      FIG.  397.  —  Section  of  a  Telephone 

wire  O.     An  iron  disk  D  is 

mounted  so  as  to  vibrate  freely  close  to  the  end  of  the 

magnet. 


458 


A   HIGH   SCHOOL   COURSE  IN   PHYSICS 


If  a  person  speaks  into  the  receiver,  the  sound  waves 
set  the  disk  in  vibration.  Each  vibration  of  the  disk 
changes  the  number  of  magnetic  lines  of  force  through 
the  coils  of  wire  and  thus  induces  a  current  whose  nature 
depends  entirely  upon  the  loudness,  pitch,  and  quality  of 
the  sound.  In  this  manner  a  pulsating  current  is  sent 
over  the  line  to  a  similar  instrument  at  the  distant  station. 
When  the  current  generated  at  the 
first  station  flows  in  such  a  direction 
as  to  strengthen  the  magnet  at  the 
second  one,  the  disk  is  drawn  in; 
when  it  flows  in  the  opposite  direc- 
FIG.  398.  — A  Recent  Type  non  the  magnet  is  weakened  and  the 

of  Receiver.  y  , 

disk  released.     Thus   the  vibrations 

at  the  first  station  are  reproduced  on  the  instrument  at  the 
second.     A  modern  receiver  is  shown  in  Fig.  398. 

473.  The  Transmitter.  —  The  telephone  receiver  just 
described  is  not  sufficiently  powerful  when  used  as  a  trans- 
mitter, or  sender,  of  speech ;  but  it  is  a  receiver,  or  repro- 
ducer, of  sound  of  extremely  great  sensibility.  For  this 
reason  the  transmitters  in  general  use 
are  based  on  an  entirely  different 
principle  from  that  of  the  receiver. 
The  modern  form  used  in  long-dis- 
tance telephony  consists  of  two  car- 
bon buttons  c  and  c',  Fig.  399, 
between  which  are  carbon  granules  g. 
The  metal  disk,  or  diaphragm,  D  is 
attached  to  the  button  <?.  When  the 
disk  is  set  in  vibration  by  the  sound  waves,  the  variation 
of  pressure  against  the  carbon  granules  causes  large  vari- 
ations in  the  electrical  resistance  between  the  buttons. 
Hence,  if  such  an  instrument  be  placed  in  a  battery  circuit 
and  so  connected  that  the  current  passes  through  the 


FIG.  399.  —  The  Trans- 
mitter. 


ELECTRO-MAGNETIC   INDUCTION 


459 


granules  from  c  to  c\  the  strength  of  the  current  changes 
precisely  in  accordance  with  the  vibrations  of  the  diaphragm. 
474.  A  Long  Distance  Telephone  System.  —  In  all  tele- 
phones operating  with  a  local  battery,  the  connections  are 
made  as  shown  in  Fig.  400.  The  current  from  the  local 


Line  Wire 


FIG.  400.  —  A  Long  Distance  Telephone  System. 

battery  B  is  led  through  the  transmitter  and  the  primary 
of  a  small  induction  coil  back  to  the  battery.  The  main 
line  contains  at  each  station  the  secondary  of  the  induction 
coil  and  the  receiver.  Two  wires  are  generally  used  to 
connect  the  two  stations,  although  one  may  be  replaced  by 
the  earth. 

When  a  person  speaks  into  the  transmitter,  the  vibration 
of  the  diaphragm  changes  the  pressure  at  the  contact 
points  of  the  carbon  granules  which  Conduct  the  current 
flowing  through  the  primary  coil.  When  the  diaphragm 
is  forced  in,  the  resistance  of  the  transmitter  is  lowered, 
and  a  comparatively  large  current  flows ;  when  it  moves 
outward,  the  current  is  reduced.  These  changes  in  the 
primary  coil  induce  currents  in  the  secondary  which  pass  out 
over  the  line  and  set  up  vibrations  in  the  receiver  at  the  dis- 
tant station. 

For  the  purpose  of  calling  attention,  an  electric  bell  is 
placed  at  each  station.  When  the  receiver  is  lifted  from 
the  hook  upon  which  it  hangs,  the  bells  are  disconnected 
from  the  line,  and  the  connections  are  made  as  shown  in 


460          A   HIGH   SCHOOL   COURSE   IN   PHYSICS 

Fig.  400.     The  downward  motion  of  the  hook  restores  the 
bell  to  the  line  when  the  receiver  is  hung  up. 

In  cities  and  villages  the  telephones  are  all  connected 
with  a  central  exchange,  where  the  operator  upon  request 
connects  the  line  from  any  instrument  with  the  line  lead- 
ing to  any  other.  Such  exchanges  can  now  be  found  in 
even  the  small  towns  and  will  serve  to  show  to  the  student 
of  electricity  one  of  the  ways  in  which  the  study  of  physi- 
cal principles  has  contributed  to  the  prosperity  and  con- 
venience of  mankind. 

EXERCISES 

1.  Compute  the  resistance  of  an  iron  telegraph  line  150  mi.  long, 
the  size  of  the  wire  being  0.1  in.,  the  line  containing  also  10  relays  of 
150  ohms  each.     Allow  nothing  for  the  earth  connections. 

2.  A  telegraph  wire  offers  a  resistance  of  35  ohms  per  mile.     If  the 
line  contains  five  150-ohm  instruments,  what  current  will  be  produced 
by  30  Daniell  cells  of  2  ohms  each  in  a  line  80  mi.  long?    Do  you 
think  this  current  would  operate  a  sounder  ? 

3.  Connect  a  telephone  receiver  with  the  terminals  of  a  very  sensi- 
tive galvanometer  and  press  in  on  the  diaphragm.     A  deflection  will 
be  produced.     Why?     Release  the  diaphragm.     A  contrary  current 
will  be  obtained.     Why? 

4.  If  the  telephone  receiver  can  be  used  as  a  transmitter  and  re- 
quires no  battery  in  its  operation,  why  is  it  not  so  used? 


SUMMARY 

1.  Currents  of  electricity  may  be  produced  by  induction 
by  increasing  or  decreasing  the  number  of  magnetic  lines 
of  force  which  thread  through  a  coil  of  wire  if  it  is  in  a 
"  closed  circuit."     The  induced  E.  M.  F.  is  proportional  to 
the  rate  of  change  in  the  number  of  lines  (§§  446  and  447). 

2.  The  direction  of  an  induced  current  is  always  such 
as  to  produce  a  field  that  tends  to  prevent  a  change  in  the 
number  of  lines  of  force  through  the  coil  (§  448). 


ELECTRO-MAGNETIC   INDUCTION  461 

3.  When  an  electrical  conductor  cuts  magnetic  lines  of 
force,  an  E.  M.  F.  is  induced  within  it  (§  450). 

4.  Currents  of  electricity  are  produced  on  a  large  scale 
by  making  use  of  the  dynamo,  which  depends  for  its  action 
on  the  principles  of  electro-magnetic  induction.     Dynamos 
are  alternators  or  direct- cur  rent  dynamos  according  as  they 
produce  alternating  or  direct  currents  (§§  451  to  458). 

5.  Dynamos  are  series-wound,  shunt-wound,  or  compound- 
wound,  depending  on  the  manner  in  which  the  field  cores 
are  excited  (§  457). 

6.  The  direct-current  motor  depends  upon  the  tendency 
of  a  conductor  carrying  an  electric  current  to  move  in  a 
magnetic  field.     Motors  are  used  wherever  electric  power 
is   to   be  converted  into  mechanical   power   for  moving 
machinery  (§  459). 

7.  Alternating-current  transformers  are  used  to  convert 
alternating  currents  of  low  potential  into  alternating  cur- 
rents   of   high   potential   and   vice  versa.     By    their  use 
unsafe  currents  of  many  thousand  volts  can  be  reduced  to 
safe  ones  for  domestic  and  commercial  use  (§§  461  to  464). 

8.  Electric   power   is  sold  by  the  kilowatt-hour.     The 
number  of  kilowatt-hours  consumed  is  given  by  the  equa- 
tion 

kilowatt-hours  =  0.001  EC  x  hours  (§  465). 

9.  The  telegraph  and  telephone  employ  the  magnetic 
effect   of  the  electric   current  in   transmitting   messages 
(§§  468  to  474). 


CHAPTER   XXI 


RADIATIONS 

1.  ELECTRO-MAGNETIC   WAVES 

475.  An  Electrical  Discharge  is  Oscillatory.  — When  an 
electric  spark  jumps  across  a  short  gap,  it  appears  to  be 
only  a  single  flash.  The  eye  is  incapable  of  determining 
whether  or  not  this  is  actually  the  case.  The  following 
experiment  may  be  used  to  investigate  the  nature  of  such 
a  discharge. 

Bend  2  or  3  feet  of  wire  into  a  hoop  R,  Fig.  401,  leaving  a  gap  of 
about  1  millimeter  at  S.  Connect  the  hoop  in  series  with  a  Leyden 
jar  L  and  the  spark  gap  of  an  induction  coil 
7  as  shown.  If  the  induction  coil  be  now  put 
into  operation,  sparks  will  be  observed  to  pass 
across  S  every  time  they  jump  across  the  gap 
at  /. 

The  spark  at  S  indicates  that  the  air 
gap  offers  a  better  path  than  the  metal 
loop  STR.  But  we  know  that  the  re- 
sistance of  the  wire  STR  is  but  a  frac- 
tion of  an  ohm,  while  that  of  the  air 
at  S  is  perhaps  millions  of  ohms.  The 
discharge  through  the  loop,  therefore, 
must  meet  with  some  impedance  other 
than  that  which  would  be  encountered  by  a  direct  current. 
From  this  and  other  effects  we  are  led  to  infer  that  the 
discharge  at  I  is  a  rapid  surging  of  electricity  back  and 
forth,  but  one  which  lasts  only  for  a  fraction  of  a  second. 
At  the  first  rush  of  current  in  the  ring  a  magnetic  field  is 

462 


Induction 
Coil 


FIG.  401.  —  Illustrat- 
ing an  Effect  of 
an  Electric  Dis- 
charge. 


RADIATIONS 


463 


FIG.  402.— Result  Obtained  by  Photo- 
graphing an  Electric  Spark. 


suddenly'set  up  within  the  loop.  This  sudden  magnetic 
change  induces,  according  to  Lenz's  law  (§  448),  an  op- 
posing E.  M.  F.  which  effec- 
tually prevents  the  flow  of  a 
large  portion  of  the  current 
in  the  loop.  Likewise  the 
sudden  reversal  of  the  dis- 
charge reverses  the  magnetic 
lines  and  again  induces  an  opposing  E.  M.  F.  in  the  loop. 
Hence  the  greater  portion  of  the  discharge  finds  a  better 

outlet  through  the  gap  iS. 
Stronger  proof  of  the  oscillat- 
ing nature  of  a  discharge  has 
been  obtained  by  photograph- 
ing a  spark  by  the  help  of  a 
revolving  mirror.  The  result 
is  shown  in  Fig.  402.  The 
period  of  oscillation  has  been 
shown  by  this  method  to  be 
'of  the  order  of  a  millionth  of  a  second.  As  a  rule  the 
oscillations  subside  very  quickly,  as  shown  by  the  curve 
in  Fig.  403. 

476.  Electro-magnetic  Waves  in  the  Ether.  —  In  1888 
Hertz  1  of  Germany  showed  that  each  electrical  oscillation 
that  occurs  when  a  spark  passes  across  an  air  gap  pro- 
duces a  disturbance  in  the  surrounding  ether  which  is 
propagated  outward  in  wave  form  in  much  the  same 
manner  as  water  waves  move  outward  from  a  falling 
pebble,  or  sound  waves  from  a  vibrating  bell.  These 
ether  waves  are  called  electro-magnetic  waves.  Inasmuch 
as  light  itself  is  propagated  in  wave  form  in  the  ether, 
it  might  be  inferred  that  the  speed  of  the  two  should  be 
the  same.  Such  has  been  found  to  be  the  case. 


FIG.  403.  —  Diagram  of  a  Spark 


1  See  portrait  facing  page  464.     See  also  Kelvin,  frontispiece. 


464         A   HIGH   SCHOOL   COURSE   IN   PHYSICS 

477.  Detection  of  Electro -magnetic  Waves.  —  Tne  process 
of  transmitting  messages  by  means  of  electro-magnetic 
ether  waves  is  dependent  on  the  detection  of  such  waves 
at  the  receiving  station.     One  method  that  can  be  em- 
ployed for  this  purpose  is  illustrated  by  the  following 
experiment. 

Bend  a  glass  tube  about  5  centimeters  long  and  3  or  4  millimeters 
in  diameter  as  shown  in  Fig.  404.     Place  a  few  coarse  iron  filings  in 
the  bend  and  introduce  a  globule  of  clean  mercury  into  each  end  of 
the  tube  as  shown  at  A  and  B.     Insert  small 
iron  wires  into  the  mercury  and   connect  the 
device  in  series  with  a  cell  and  an  ammeter  or 
galvanometer.     No   deflection  should   be  pro- 
FIG.  404.  — A  Coherer     duced.     Now  cause  a  spark  to  jump  a  short  air- 
gap   several  feet   away  by  using  an  induction 

coil,  an  influence  machine  (§  371),  or  a  Ley  den  jar.  A  deflection 
will  be  observed  at  once.  If  the  tube  of  filings  be  now  tapped  lightly, 
the  circuit  is  again  broken,  and  the  experiment  may  be  repeated. 

The  tube  of  metal  filings  used  as  a  detector  of  electro- 
magnetic waves  is  called  a  coherer.  Ordinarily  the  filings 
offer  a  large  resistance  to  the  flow  of  electricity  through 
the  many  points  of  contact  between  the  several  pieces. 
The  waves  emitted  by  the  oscillatory  discharge  at  the 
spark  gap  cause  the  filings  to  cling  together,  and  the 
resistance  of  the  tube  immediately  falls  to  a  few  ohms. 
A  tap  or  jar  breaks  the  filings  apart,  and  the  resistance 
rises  again  to  its  former  value. 

478.  Wireless  Telegraphy.  —  The  possibility  of  sending 
out  electro-magnetic  waves  from  an  induction  coil  and  of 
detecting  them  by  a  coherer  has  led  to  the  transmission 
of  messages  by  the   so-called  wireless  telegraph  systems. 
A  plan  of  a  simple  sending  and  receiving  equipment  is 
shown  in  Figs.  405  and  406. 

Connect  the  coherer  described  in  the  preceding  section  in  series 
with  the  electro -magnet  of  a  relay  R  and  a  cell  C,  Fig.  406.  In  the 


HEINRICH    HERTZ    (1857-1894) 

About  the  middle  of  the  last  century  Maxwell  advanced  the  idea 
that  waves  of  light  are  electro-magnetic  in  character.  If  this  were 
true  of  light,  then  it  would  also  be  true  of  radiant  heat.  In  1888 
Hertz  of  Germany  succeeded  in  demonstrating  experimentally  the 
truth  of  this  assumption.  During  the  discharge  of  electricity  between 
two  polished  knobs,  so-called  electro-magnetic  waves  are  radiated  into 
space.  Hertz  was  able  to  detect  these  waves  and  to  reflect,  refract, 
and  polarize  them  and  to  cause  interference  to  take  place  between 
them.  He  also  measured  the  velocity  with  which  they  are  propa- 
gated through  space,  and  found  it  to  be  equal  to  the  velocity  of 
light.  Thus  the  hypothesis  of  Maxwell  was  placed  upon  an  experi- 
mental basis,  and  the  way  opened  for  long-distance  communication 
between  stations  without  the  necessity  of  connecting  wires.  The 
practical  value  of  Hertz's  results  can  hardly  be  overestimated. 
Among  others  who  have  contributed  greatly  to  the  knowledge  of 
electro-magnetic  waves  may  be  mentioned  Sir  Oliver  Lodge  of  Eng- 
land, Righi  of  Bologna,  and  Branly  of  Paris. 

In  1880  Hertz  was  made  assistant  to  Helmholtz  at  Berlin.  In 
1885  he  became  professor  of  physics  in  the  Technical  High  School 
at  Karlsruhe.  It  was  in  the  latter  place  that  his  epoch-making 
experiments  were  first  performed.  In  1889  he  was  elected  professor 
of  physics  at  Bonn,  where  he  died  at  the  age  of  thirty-seven  years. 
Electro-magnetic  waves  are  called  Hertzian  waves  in  his  honor. 


RADIATIONS 


465 


H'l'l'- 


make-and-break  circuit  of  the  relay  connect  an  electric  bell  B  (§  411) 
and  a  cell  C'.  The  bell  is  so  placed  that  its  hammer  strikes  the  co- 
herer D  whenever  it  is  set  in  vibration.  One  end  of  the  coherer 
should  be  connected  to  the  earth  and  the  other  joined  to  a  high  aerial 
wire.  For  short  distances  the  aerial  wire  may  be  very  short  or  left 
off  altogether.  At  the  sending  station  (Fig.  405),  simply  connect 
one  side  of  a  spark  gap  of  an  induction  coil  with  the  earth  and 
join  the  other  side  with  an  aerial  conductor  which  is  merely  a 
rod  or  wire,  one  end  of  which  is  lifted  some  distance  above 
the  apparatus.  A  key  K  should  be  placed  in  the  circuit  of 
the  primary  coil.  Closing  the  key  and  thus  producing  sparks 
at  S  causes  the  bell  to  ring  at  the  receiving  station  until  the 
circuit  at  the  sending  station  is  broken. 

When  sparks  pass  at  the  so- 
called  oscillator  S,  electro-mag- 
netic waves  are  emitted  which 
affect  the  coherer  at  the  receiving 
station,  causing  the  filings  to  con- 
duct as  shown  in  §  477.  The  cur- 


Earth 

FIG.  405.  —  Apparatus  for 
Sending  Wireless  Tele- 
graph Signals. 


rent  from  the  cell  0  then  flows 
through  the  coherer  and  the 
electro-magnet  of  the  relay 
R.  The  relay  closes  at  A  the 
circuit  through  the  bell  which 
is  rung  by  the  current  from 
the  cell  0'.  The  filings  con- 
tinue to  cohere  until  sparks 
cease  to  pass  at  $  when  the 
taps  of  the  bell  hammer  jar  FIQ  m_A  Receiving  Station  for 

them    apart.        The     circuit  Wireless  Messages. 

31 


Earth 


466 


A  HIGH   SCHOOL   COURSE   IN   PHYSICS 


through  the  relay  is  thus  broken,  which  in  turn  opens 
the  bell  circuit  at  A.  Another  spark  at  S  again  causes 
D  to  conduct  and  the  bell  to  ring.  Many  other  forms 
of  receiving  devices  are  in  common  use. 

Systems  of  wireless  telegraphy  have  reached  such  a 
state  of  development  that  ocean-going  vessels  are,  as  a 
rule,  at  all  times  in  communication  with  land  stations  or 
with  one  another.  Passengers  on  sinking  vessels  have 
thus  been  rescued  by  timely  assistance  obtained  through 
wireless  messages  which  were  received  at  stations  many 
miles  away.  Naval  fleets  equipped  with  a  good  system 
may  be  kept  continually  informed  by  the  controlling 
department  of  the  government,  and  the  department  may 
be  kept  acquainted  with  every  movement  of  the  fleet. 
Even  transatlantic  messages  are  now  transmitted  without 
the  use  of  wires  or  cables. 

2.    CONDUCTION   OF   GASES 

479.  Conduction  through  Vacuum  Tubes.  —  Connect  the  ter- 
minals of  an  induction  coil  with  electrodes  A  and  B,  Fig.  407,  which 

are  sealed  in  the  ends  of  a  glass 
tube  2  or  3  feet  in  length.  Con- 
nect the  tube  with  an  air  pump 
and  put  the  coil  in  operation. 
Sparks  will  jump  across  the  gap  S 
because  it  is  the  better  path. 
Put  the  air  pump  in  action  and 
exhaust  the  tube  while  the  coil 
is  running.  When  the  pressure 
has  been  sufficiently  reduced,  the 
discharge  will  begin  to  take  place 
through  the  long  exhausted  tube 
rather  than  over  the  shorter  path  through  the  air  at  S. 

When  the  exhaustion  of  the  gas  has  been  carried  much 
farther,  there  may  be  seen  a  radiation  from  the  cathode 
proceeding  in  straight  lines  and  traceable  by  a  slight 


FIG.  407.  —  Discharge  through  a  Par- 
tial Vacuum. 


SIR    WILLIAM    CROOKES    (1832-         ) 


The  discharge  of  electricity  through 
partially  exhausted  tubes  has  been  a  sub- 
ject of  much  research  since  the  middle  of 
the  last  century.  In  1853  Masson  of  Paris 
discharged  electricity  from  a  large  induc- 
tion coil  through  a  Torricellian  vacuum. 
Later  Geissler,  a  German,  constructed  ex- 
cellent tubes  containing  small  amounts  of 
different  gases,  which  became  famous  on 
account  of  the  great  beauty  of  color  mani- 
fest upon  discharging  electricity  through 
them.  Crookes  began  his  experiments  in 
1873  with  tubes  in  which  the  exhaustion 
was  carried  to  a  high  degree.  In  these  he 
showed  that  so-called  "  radiant  matter "  is 
thrown  out  from  the  electrodes  in  straight  lines,  casts  shadows  when 
intercepted  by  solids,  and  is  capable  of  producing  mechanical  effects 
when  brought  into  collision  with  light  movable  vanes.  Furthermore, 
he  showed  that  the  stream  of  particles  is  deflected  from  a  straight 
line  by  a  magnet. 


WILHELM  KONRAD  RONTGEN  (1845-    ) 


The  most  striking  phenomena  accom- 
panying discharges  in  highly  exhausted 
tubes  were  discovered  by  Professor  Ront- 
gen  at  Wiirzburg,  Germany,  in  1895.  A 
Crookes  tube  was  left  in  operation  on  a 
table.  Beneath  it  was  a  book  containing 
a  key  as  a  bookmark;  below  the  book  was 
a  photographic  plate  in  a  plate  holder. 
Later,  on  using  the  plate  in  a  camera,  and 
developing  it  in  the  usual  manner,  a  well- 
defined  shadow  of  the  key  became  visible 
upon  it.  Further  experimentation  showed 
the  discoverer  that  he  had  found  a  new 
kind  of  radiation  which  he  named  X-rays. 
The  important  ends  attained  by  the  use  of 
X-rays  in  the  practice  of  medicine  and  surgery  will  always  serve  to 
keep  the  name  of  Rontgen  before  the  public. 


OF  THE 

UNIVERSITY 

£AUFO! 


RADIATIONS  467 

luminescence  occasioned  in  the  gas  remaining  in  the  tube. 
Where  these  rays  fall  upon  the  glass,  it  is  made  warm 
and  luminous.  Ordinary  glass  glows  with  a  soft  greenish 
yellow  light ;  and  a  solid,  as  marble,  placed  in  the  path 
of  the  radiations  will  glow  with  a  characteristic  color. 
These  radiations  are  known  as  cathode  rays.  Such  highly 
exhausted  and  hermetically  sealed  tubes  are  known  as 
Crooked l  tubes. 

480.  Cathode  Rays.  —  Cathode  rays   are   characterized 
by  three  main  properties,  viz.  (1)  they  are  deflected  from 
a  straight  line  when  caused  to  pass  through  an  electric  or 
a  magnetic  field,  (2)  they  convey  a  negative   charge  of 
electricity  to  the  object  upon  which  they  fall,  and  (3)  they 
raise  the  temperature  of  any  solid  object  which  obstructs 
their  path.     The  inference  is,  therefore,  that  cathode  rays 
are  swiftly  moving  particles  charged  with  negative  electricity. 
The  velocity  with  which  the   particles   move  sometimes 
reaches   the   enormous   value   of   over   50,000   miles   per 
second,  —  nearly  one  third  of  the  velocity  of   light.     A 
continuous  stream  of  such  particles,  all  of  which  carry 
negative  charges,  is  equivalent  to  a  current  of  electricity ; 
hence  their  deviation  in  passing  through  a  magnetic  field. 

481.  X-Rays.  —  The  most  important  application  of  the 
action  of  cathode  rays  is  employed  in  the  production  of 
the    so-called    X-rays, 

which  were  discovered 
in  1895  by  Rontgen,1  a 
German  physicist.  A 
special  vacuum  tube  of 
the  form  shown  in  Fig. 

,  FiG.408.-AnX-RayTube. 

408  is  used  for  this  pur- 
pose.    This  is  connected  with  the  secondary  of  an  induc- 
tion coil  in  such  a  manner  that  (7,  a  concave  electrode,  is 
1  See  portrait  facing  page  466. 


468         A  HIGH   SCHOOL   COURSE   IN   PHYSICS 

the  cathode,  and  P  the  anode.  Since  the  cathode  particles 
are  sent  off  at  right  angles  to  the  surface  which  they  leave, 
they  are  focused  against  the  solid  piece  of  platinum  P 
placed  near  the  center  of  the  tube.  Accompanying  the 
impact  of  the  cathode  rays  against  P  appears  a  new  radia- 
tion of  an  entirely  different  character.  It  is  to  these  that 
the  name  of  X-rays,  or  Rontgen  rays,  has  been  given. 

482.  Properties  of  X-Rays.  —  The  properties  which  give 
to  X-rays  their  great  practical  utility  is  their  power  (1)  to 
penetrate  bodies  of  matter  that  are  opaque,  and  (2)  to 
make  a  permanent  impression  upon  photographic  plates. 
Substances  are  not  all  equally  transparent  to   the  rays ; 
e.g.  they  more  readily  penetrate  a  thick  book  than  a  thin 
coin  ;  and  while  flesh  is  quite  transparent,  bones  are  more 
or  less  opaque.     Hence,  if  the  hand  be  held  between  an 
X-ray  tube  and  a  photographic  plate,  the  bones  will  shield 
the  plate  more  than  the  flesh  and  thus  produce  shadows. 
Fig.  409  shows  an  X-ray  picture  of  a  broken  wrist,  and 
the  same  wrist  is  shown  in  Fig.  410  after  having  healed. 

3.    RADIO-ACTIVITY 

483.  Radio-active  Substances.  —Place  a  gas  mantle  that  has 
been  pressed  out  flat  upon  a  photographic  plate  and  inclose  it  in  a 
light-proof  box.     After  three  or  four  weeks  develop  the  plate  in  the 
usual  way.     A  distinct  image  of  the  fabric  of  the  mantle  will  become 
visible.     Such  a  plate  is  shown  in  Fig.  411. 

Similar  experiments  may  be  made  with  compounds  con- 
taining uranium,  especially  by  using  the  mineral  uraninite, 
or  pitchblende.  These  experiments  show  that  certain  sub- 
stances spontaneously  emit  a  kind  of  radiation  capable  of 
affecting  a  photographic  plate.  Such  substances  are  said 
to  be  radio-active,  or  to  possess  the  property  of  radio-activ- 
ity. The  radio-active  element  contained  in  a  gas  mantle 
is  thorium.  The  element  radium  is  remarkable  for  its 


FIG.  409.— X-Ray  Picture  Showing 
the  Fractured  Bones  of  a  Wrist. 


FIG.  410.— X-Ray  Picture  of  the 
Wrist  Shown  in  Fig.  409,  after  the 
Bones  had  Knitted. 


FIG.   411. — The  Radio-active  Effect  of  a  Portion  of  a  Gas  Mantle  on 
Photographic  Plate. 


RADIATIONS  469 

intense  radio-activity.  The  discovery  of  this  element  by 
Monsieur  and  Madame  Curie l  of  Paris,  in  1898,  resulted 
from  the  fact  that  its  presence  in  microscopic  quantities 
in  tons  of  the  mineral  from  which  it  was  taken  was  de- 
tected by  its  exceptionally  large  radio-active  effects. 

484.  Radio-Activity.  —  The  property  of  radio-activity 
was  discovered  by  Henri  Becquerel 2  of  Paris  in  1896.     In 
honor  of  the  discoverer  the  radiations  emitted  by  radio- 
active substances  are  often  called  Itecquerel   rays.     Ex- 
perimental researches  have  shown  that  Becquerel  rays  are 
of   a  complex  nature.     They   consist   (1)   of   negatively 
charged    particles    called    beta    rays,    (2)    of    positively 
charged   particles   called    alpha   rays,  and   (3)    of   radia- 
tions   resembling    X-rays    in    nature,   which    are    called 
gamma  rays. 

Becquerel  rays  are  detected  not  only  by  their  power  to 
affect  a  photographic  plate  as  shown  in  §  483,  but  also  by 
their  power  to  discharge  electrified  bodies  near  which  they 
are  placed.  This  effect  is  due  to  the  fact  that  the  air 
surrounding  a  radio-active  body  is  rendered  a  conductor 
of  electricity  by  the  radiations  emitted.  Furthermore, 
if  minute  crystals  of  zinc  sulphide  be  placed  in  the 
immediate  neighborhood  of  an  extremely  small  quantity 
of  radium,  they  show  intermittent  flashes  of  light  as  they 
are  bombarded  by  the  alpha  particles  expelled  by  the 
radium. 

485.  Electrons.  —  Beta  rays,  or  the  negatively  charged 
particles  emitted  by  a  radio-active  substance,  are  called 
negative  electrons.     Negative  electrons  are  separated  from 
alpha  and  gamma  rays  on  passing  through  a  strong  electric 
or  magnetic  field  on  account  of  the  difference  in  the  kind  of 
charge  carried  by  them.     The  path  of  the  negative  elec- 
trons is  curved  in  one  direction  by  the  field,  while  that  of 

1  See  portrait  facing  page  470. 


470         A  HIGH   SCHOOL  COURSE   IN   PHYSICS 

the  alpha   rays  is   bent  in  the  opposite    direction.     The 
gamma  rays  remain  unchanged  in  direction. 

Knowledge  of  electrons  began  with  the  discovery  by  Sir 
J.  J.  Thompson  of  England  that  the  cathode  rays  pro- 
duced in  a  vacuum  tube  (§  480)  consist  of  swiftly  moving 
particles  charged  with  electricity.  These  particles  have 
since  been  found  to  be  identical  with  the  negative  elec- 
trons emitted  by  a  radio-active  substance.  The  speed 
attained  by  the  electrons  in  a  cathode  tube  is  about 
62,000  miles  per  second,  but  the  electrons  emitted  by 
radium  move  with  speeds  that  reach  as  high  as  165,000 
miles  per  second,  which  is  about  -f$  the  velocity  of 
light. 

The  mass  of  an  electron  has  been  ascertained  and  found 
to  be  about  yyVir  ^ne  mass  °f  an  atom  of  hydrogen.  Dif- 
ferences in  the  electrical,  magnetic,  and  other  properties 
of  matter  are  attributed  to  variations  in  the  arrangements 
and  movements  of  the  electrons  associated  with  the  mole- 
cules. For  example,  in  electrical  conductors  the  electrons 
are  less  firmly  attached  to  the  molecules  than  they  are  in 
non-conductors,  and  are  consequently  moved  readily  by 
potential  differences.  This  motion  of  electrons  through 
a  conductor  constitutes  an  electric  current. 

486.  Alpha  Particles.  —  The  alpha  rays  emitted  by  a 
radio-active  substance  are  of  atomic  size  and  have  been 
found  to  be  charged  with  positive  electricity.  Experi- 
ments have  also  shown  that  they  are  atoms  of  the  well- 
known  gas  helium,  and  that  their  velocity  may  reach  as  high 
as  -^-Q  the  speed  of  light.  The  energy  that  is  developed 
by  a  radio-active  body  is  due  mainly  to  the  alpha  particles 
which  it  hurls  outward  with  this  enormous  rate  of  motion. 
It  is  estimated  that  the  total  amount  of  energy  that  can 
be  given  off  by  a  gram  of  radium  is  about  equal  to  that 
developed  by  the  combustion  of  a  ton  of  coal. 


ANTOINE    HENRI    BECQUEREL    (1852-1908) 


A  new  epoch  in  the  theory  of  matter  was 
inaugurated  by  the  discovery  of  radio- 
activity in  1896  by  Becquerel  of  Paris.  It 
had  long  been  known  that  compounds  con- 
taining the  element  uranium  would  produce 
an  effect  upon  photographic  plates  in  the 
dark,  This  singular  action  had  been  attrib- 
uted merely  to  their  property  of  phosphor- 
escence. Becquerel  proved  that  all  com- 
pounds containing  uranium  act  similarly  on 
plates,  even  those  which  are  not  phosphor- 
escent. This  phenomenon  corresponds  to  a 
continuous  emission  of  energy,  for  which 
the  old  view  of  matter  did  not  account. 


MADAME    CURIE    (1867-         ) 

Soon  after  the  discovery  of  radio-activ- 
ity, investigations  were  made  by  Madame 
Curie  of  Paris  and  others  in  order  to  as- 
certain if  radio-activity  were  not  a  general 
property  of  matter.  Various  compounds 
were  tested;  but  the  strange  property 
appeared  to  be  confined  to  substances 
containing  uranium  and  another  element, 
thorium.  However,  it  was  observed  by 
Madame  Curie  that  certain  pitchblendes 
(minerals  containing  oxide  of  uranium  and 
other  well-known  elements)  were  many 
times  more  radio-active  than  uranium.  It 
therefore  seemed  probable  that  an  unknown 
element  of  great  radio-activity  was  pres- 
ent in  the  mineral.  From  this  grew  up  the  celebrated  experiments  of 
Monsieur  and  Madame  Curie  which  led  to  the  discovery  of  the  re- 
markable element  radium.  Although  the  separation  of  radium  from 
minerals  is  attended  with  enormous  difficulties  on  account  of  the 
small  quantity  which  they  contain,  enough  of  it  has  been  secured  for 
experimental  research  in  the  laboratories  of  the  world  to  lead  to  the 
generalizations  in  regard  to  the  constitution  of  matter  contained  in 
the  concluding  sections  of  this  book. 


RADIATIONS  471 

487.  Disintegration  of  Matter. — The  emission  of  elec- 
trons and  of  positively  charged  particles  which  are  compar- 
able in  mass  with  atoms  of  hydrogen  leads  at  once  to  the 
conclusion  that  a  radio-active  substance  must  be  continu- 
ally experiencing  a  molecular  change.     Although  a  gram 
of  radium  develops  hourly  an  amount  of  heat  equal  to  100 
calories,  its  transformation  is  so  slow  that  nearly  13  centu- 
ries would  elapse  before  one  half  its  mass  would  suffer  a 
change  in  its  nature.     Experiments  have  shown  that  it  is 
along  with  this  transformation  that  the  element  helium  is 
produced.     Thus  the  radio-active  elements  are  not  perma- 
nent, but  by  the  emission  of  electrons  and  alpha  particles 
they  are  being  constantly  transformed  into  elements  of 
smaller  atomic  weights.      Uranium,  for  example,  is  sup- 
posed to  be  breaking  up  and  forming  for  one  of  its  products 
the  element  radium  ;  radium  in  turn  forms  so-called  radium 
emanation,  and  so  on  through  a  series.     It  is  while  these 
transformations  are  going  on  that  the  alpha,  beta,  and 
gamma  rays  are  emitted. 

According  to  the  disintegration  theory  of  matter  ad- 
vanced by  Rutherford  and  Soddy,  the  atoms  of  radio-active 
substances  are  unstable  systems  which  break  up  spontane- 
ously with  explosive  violence,  expelling  a  small  portion 
of  the  fractured  atom  with  great  velocity.  The  remaining 
portion  of  each  atom  forms  a  new  system  of  a  smaller 
atomic  weight  and  possessing  properties  differing  from 
those  of  the  atoms  of  the  parent  substance.  This  new 
substance  may  be  unstable  also  and  undergo  an  atomic 
change  similar  to  the  preceding.  The  process  may  con- 
tinue by  stages  until  at  last  a  stable  form  is  finally 
attained. 

488.  The  Domain  and  Future  of  Physics.  —  The  study 
of  Physics  is  primarily  an  investigation  of  environment  to 
the  end  that  the  knowledge  obtained  shall  be  conducive  to 


472         A    HIGH   SCHOOL  COURSE   IN   PHYSICS 

the  comfort  of  mankind,  arid  lead  to  an  increase  of  man's 
power  in  his  range  of  action.  If  this  were  all,  the  pursuit 
of  physical  science  would  be  amply  justified,  for  the  field 
is  a  broad  one,  and  the  results  to  which  the  study  has  led 
are  overwhelming.  It  has  opened  the  entire  world  to 
the  traveler,  it  has  brought  the  West  within  speaking 
distance  of  the  East,  it  has  overthrown  the  dangers  and 
solitude  of  the  sea,  it  has  brought  distant  worlds  within  our 
range  of  vision  and  exposed  the  sources  of  disease,  it  re- 
veals the  secret  of  aerial  flight,  and,  now,  at  the  present 
rate  of  advancement  man  may  soon  acquire  that  knowledge 
which  the  human  mind  has  long  sought  to  secure  in  answer 
to  the  all-important  question,  "  What  is  matter  ?  " 


INDEX 


Absolute  temperature,  222. 

Absolute  units  of  energy,  52. 

Absolute  units  of  force,  30,  31. 

Absorption,  of  heat,  253;  of  ra- 
diant energy,  253;  selective, 
254;  spectra,  327. 

Accelerated  motion,  17. 

Acceleration,  16;  due  to  gravity, 
70;  centripetal,  45. 

Activity,  53. 

Adhesion,  130. 

Aeronauts,  altitude  reached  by, 
153. 

Aeroplane,  43. 

Air,  buoyancy  of,  154;  compres- 
sibility of,  147;  density  of, 
138;  pressure  of,  139,  155; 
variations  in  pressure  of,  141. 

Air  brake,  161. 

Alpha  particles,  470. 

Alternating  currents,  434. 

Alternator,  435. 

Altitude,  effect  of,  on  barometer, 
153;  effect  of,  on  boiling  point, 
242. 

Amalgamation  of  zinc,  379. 

Ammeter,  404. 

Ampere,  sketch  and  portrait  of, 
facing  406. 

Amplitude,  74;  effect  of,  on  in- 
tensity of  sound,  174. 

Aneroid  barometer,  143. 

Anode,  396. 

Antinode,  194,  201. 

Arc,  lamp,  451;  light,  450;  elec- 
tric, 450;  enclosed,  451;  flam- 
ing, 451;  open,  451. 

Archimedes,  119;  principle  of, 
119. 


Armature,  392,  434;  drum,  438; 
Gramme  ring,  437. 

Artesian  wells,   114. 

Athermanous  substances,  255. 

Atmosphere,  as  unit  of  pressure, 
142;  density  of,  138,  152,  153; 
humidity  of,  239;  pressure  of, 
139. 

Atoms,  471. 

Attraction,  electrical,  335;  mag- 
netic, 361;  molecular,  130. 

Balance,  10;  spring,  10. 

Balloon,  155. 

Barometer,    142;     aneroid,    143; 

self-recording,   143;   utility  of, 

144. 

Battery,   storage,   399. 
Beats,   190;    law  of,  191. 
Becquerel,  sketch  and  portrait  of, 

facing  470. 

Bell,  diving,  162;  electric,  392. 
Binocular  vision,  317. 
Bodies,  falling,  70;  thrown,  72. 
Boiling,  241;   laws  of,  242. 
Boiling  point,  of  liquids,  243 ;  on 

thermometers,  213. 
Boyle's  law,  149. 
Bright-line  spectra,  328. 
Bunsen  photometer,  276. 
Buoyancy,  of  air,  154 ;  of  liquids, 

119. 

Caisson,  pneumatic,  162. 

Caloric,  257. 

Calorie,     denned,     225;     electric 

equivalent  of,  415;  mechanical 

equivalent  of,  258. 
Camera,  312. 


473 


474 


A  HIGH   SCHOOL  COURSE  IN   PHYSICS 


Candle  power,  defined,  276;  of 
lights,  275. 

Capacity,  electric,  348. 

Capillarity,  132;  in  soil,  137;  in 
tubes,  133. 

Capstan,  95. 

Cathode,  396. 

Cathode  rays,  467. 

Cell,  chemical  action  in,  377 ; 
Daniell,  385;  dichromate,  384; 
"dry,"  387;  gravity,  385;  Le- 
clanche",  386;  local  action  in, 
379;  storage,  399;  theory  of, 
378;  voltaic,  375. 

Cells,  in  parallel,  411;  in  series, 
410. 

Center  of  gravity,  65;  of  mass, 
65;  of  oscillation,  79;  of  per- 
cussion, 79. 

Centimeter-gram-second  system, 
6. 

Centrifugal  force,  45. 

Centripetal  acceleration,  45. 

Centripetal  force,  45,  46. 

Charges,  electrical,  335;  mixing, 
348. 

Charging,  by  contact,  338;  by  in- 
duction, 342. 

Charles,  law  of,  222. 

Chemical  effects  of  currents,  395. 

Chord,  major,  182. 

Circuit,  divided,  420;  electric, 
377. 

Coefficient  of  expansion,  of  gases, 
220;  of  liquids,  220;  of  solids, 
217. 

Coherer,  464. 

Cohesion,  130. 

Coil,  induction,  429;  magnetic 
action  of,  389. 

Cold  storage,  245. 

Color,  321;  and  wave  length, 
322;  by  dispersion,  321;  by  in- 
terference, 323;  of  films,  332; 
of  objects,  322;  of  pigments, 
324;  of  transparent  bodies, 
323;  complementary,  323. 


Commutator,  436. 

Compass,  368. 

Component  forces,  36. 

Component  motions,  21. 

Composition,  of  forces,  36;  of 
motions,  21. 

Concave  lens,  301,  303. 

Concave  mirror,  284,  287. 

Condenser,  349. 

Conduction,  of  electricity,  338; 
of  gases,  466;  of  heat,  248. 

Conductors,  of  heat,  248;  charge 
on  outside  of,  343. 

Conjugate  foci,  of  lenses,  304; 
of  mirrors,  289. 

Conservation  of  energy,  59. 

Convection,  249. 

Convex  lens,  301,  305. 

Convex  mirror,  284,  285. 

Cooling,  by  expansion,  259;  arti- 
ficial, 245. 

Couple,  41. 

Critical  angle,  297. 

Crookes,  sketch  and  portrait  of, 
facing  466. 

Crookes'  tubes,  467. 

Curie,  Mme.,  sketch  and  portrait 
of,  facing  470. 

Currents,  electric,  375;  chem- 
ical effect  of,  395;  heating 
effect  of,  394;  induced,  425; 
magnetic  effect  of,  388;  meas- 
urement of,  402;  mutual  effect 
of,  393. 

Curvature,  center  of,  284,  301. 

Curvilinear  motion,  44. 

Daniell  cell,  385. 
Declination,  369. 
Density,  10,  124;  of  air,  138; 

of  liquids,  127;  of  solids,  124; 

atmospheric,  153;  electric,  344. 
Dew  point,  240. 
Diamagnetic  substances,  362. 
Diathermanous  substances,  255. 
Diatonic  scale,  181. 
Dichromate  cell,  384. 


INDEX 


475 


Dielectric,  350. 

Dip,  magnetic,  371. 

Discharge,  electric,  351;  oscil- 
latory, 462. 

Dispersion,  321. 

Distillation,  243. 

Diver,  162. 

Drum  armature,  438. 

"Dry"  cell,  387. 

Dynamo,  432 ;  alternating-cur- 
rent, 434;  compound-wound, 
440;  rule  of,  432;  series- 
wound,  439 ;  shunt-wound, 
439;  unipolar,  440. 

Dyne,  30. 

Earth's  magnetism,  368. 

Ebullition,  237,  241;  laws  of, 
241. 

Echoes,  177. 

Eclipses,  272. 

Efficiency,  99;  of  lamps,  448, 
449;  of  simple  machines,  99. 

Elastic  force  in  gases,  148. 

Electric  bell,   392. 

Electric  car,  443. 

Electric  charge,  unit  of,  348 ;  dis- 
tribution of,  343. 

Electric  motor,  442. 

Electrical  attraction,  335. 

Electrical  circuits,  377. 

Electrical  currents,  375. 

Electrical  machines,  354. 

Electrical  repulsion,  336. 

Electrical  resistance,  405. 

Electricity,  current,  375;  static, 
335. 

Electrolysis,  396. 

Electro-magnet,  391. 

Electro-magnetic  induction,  425. 

Electromotive  force,  376;  in- 
duced, 426,  431. 

Electrons,  469. 

Electrophorus,  352. 

Electroplating,  397. 

Electroscope,  336. 

Electrotyping,  398. 


Energy,  54;  of  the  sun,  255; 
conservation  of,  59;  electric, 
413;  equation  for,  56;  heat, 
210;  kinetic,  56;  potential, 
55;  radiant,  253;  transfer- 
ences and  transformations  of, 
58. 

Engine,  gas,  261;  steam,  260; 
turbine,  263. 

English  equivalents  of  metric 
units,  5,  6,  8,  53. 

Equilibrant,  38. 

Equilibrium,  66;  neutral,  67; 
stable,  66;  unstable,  66. 

Erg,  ^ 

Ether,  252,  268,  463. 

Evaporation,  237;    laws  of,s  237. 

Expansion,  216;  of  gases,  147, 
221;  coefficient  of,  217;  un- 
equal, 218. 

Extension,  2,  4. 

Eye,   312. 

Falling  bodies,  69. 

Faraday,  sketch  and  portrait  of, 
facing  426. 

Field,  electric,  340;  magnetic, 
363. 

Field-magnet  of  dynamo,  439. 

Floating  bodies,  122. 

Floating  dry  dock,  123 

Fluids,  138. 

Focal  length,  of  lens,  302;  of 
mirror,  285. 

Focus,  conjugate,  289,  304;  prin- 
cipal, 284,  302;  real,  285;  vir- 
tual, 285,  304. 

Foot-pound,  52. 

Foot-pound-second  system,  10. 

Force,  29;  centrifugal,  45;  cen- 
tripetal, 45,  46;  component  of, 
36;  composition  of,  36;  field 
of,  340;  lines  of,  340,  363; 
moment  of,  39;  parallelogram 
of,  37;  representation  of,  35; 
resolution  of,  42;  units  of, 
30,  31. 


476 


A  HIGH   SCHOOL  COURSE  IN   PHYSICS 


Forces,  parallel,  40. 

Franklin,  sketch  and  portrait  of, 

facing  346. 

Fraunhofer  lines,  327. 
Freezing,    231;    heat    given    out 

during,  235. 

Friction,  99;  rolling,  100. 
Fundamental  tone,  194,  200. 
Fusion,  231;  of  ice,  234;  heat  of, 

233;   laws  of,  232. 

Galileo,   sketch   and  portrait  of, 

facing  70. 
Galvanometer,  382;  astatic,  403; 

d'Arsonval,  403;  tangent,  403. 
Gas  engine,  261. 
Gases,    characteristics    of,    138; 

compressibility  of,  147. 
Gilbert,  368. 

Gram,  of  force,  32;  of  mass,  7. 
Gramme  ring,  437. 
Gravitation,  law  of,  62. 
Gravity,  acceleration  due  to,  70; 

center    of,    65;    force    of,    63; 

variation  of,  70. 
Gravity  cell,  385. 
Guericke,  Otto  von,  155. 

Harmonic,  196. 

Heat,  210;  due  to  absorption  of 
radiant  energy,  255;  due  to 
electric  current,  415;  of  fusion, 
233;  of  sun,  255;  of  vaporiza- 
tion, 249;  conduction  of,  248; 
convection  of,  249;  measure- 
ment of,  225;  mechanical 
equivalent  of,  258;  produced 
by  compression,  210;  produced 
by  friction,  210;  radiation  of, 
252;  specific,  226;  unit  of, 
225. 

Heating,  by  hot  air,  250;  by  hot 
water,  251;  electrical,  394. 

Helmholtz,  sketch  and  portrait 
of,  facing  196. 

Henry,  sketch  and  portrait  of, 
facing  426. 


Hertz,  sketch  and  portrait  of, 
facing  464. 

Horse  power,  53;  electric  equiva- 
lent of,  415. 

Humidity,  240. 

Huyghens,  79,  268. 

Hydraulic  elevator,  117. 

Hydraulic  press,   112. 

Hydraulic  ram,  118. 

Hydrometers,  127. 

Hydrostatic  paradox,  109. 

Ice,  artificial,  245. 

Images,  by  double  reflection,  282 ; 
by  lenses,  305,  307;  by  plane 
mirror,  280;  by  spherical  mir- 
rors, 284;  pin-hole,  273;  real, 
287;  size  of,  310;  virtual,  287. 

Incandescent  lamp,  447. 

Incandescent  lighting,  447,  449. 

Incidence,  angle  of,  279,  293. 

Inclination  of  magnetic  needle, 
371. 

Inclined  plane,  96. 

Index  of  refraction,  294;  meas- 
ured, 295. 

Induced  charges,  342. 

Induced  magnetism,  362. 

Induction,  charging  by,  342; 
electro-magnetic,  425;  electro- 
static, 340;  magnetic,  362. 

Induction  coil,  429. 

Inertia,  29. 

Influence  machine,  354. 

Insulators,  338. 

Intensity,  of  light,  275;  of 
sound,  173. 

Interference,  of  light,  330;  of 
sound,  188. 

Intervals,  182. 

Ions,  378,  396. 

Isobars,  145. 

Isogonic  lines,  370. 

Jackscrew,  98. 
Jar,  Leyden,  351. 


INDEX 


477 


Joule,    sketch    and    portrait    of, 

facing  258. 

Joule's  equivalent,  258. 
Joule's  law,  415. 

Kelvin,  Lord,  sketch  and  por- 
trait of,  frontispiece. 

Key,  tone,  181;  telegraph,  453. 

Kilogram,  standard,  7. 

Kilogram-meter,  52. 

Kinetic  energy,  56;  equation  of, 
56. 

Kinetoscope,  317. 

Kirchoff,  328. 

Lamps,  arc,  450;  incandescent, 
447;  Nernst,  448. 

Lantern,  projection,  316. 

Laws  of  motion,  28. 

Leclanche  cell,  386. 

Lenses,  300;  achromatic,  322; 
concave,  301,  303;  converging, 
302;  convex,  301,  305;  diverg- 
ing, 303;  equation  of,  310. 

Lenz's  law,  427. 

Level,  liquid,   110. 

Lever,  88;  kinds  of,  89;  mechan- 
ical advantage  of,  89. 

Ley  den  jar,  351. 

Light,  dispersion  of,  321;  inter- 
ference of,  330;  meaning  of, 
268;  reflection  of,  278;  refrac- 
tion of,  292;  speed  of,  269. 

Lightning,  345. 

Lightning  rods,  345. 

Lines,  of  force,  340;  isogonic, 
370. 

Liquefaction,  see  Fusion. 

Liquid,  in  communicating  ves- 
sels, 110;  density  of,  127;  pres- 
sures in,  103;  pressures  trans- 
mitted by,  111;  surface  ten- 
sion of,  132;  thermal  conduc- 
tivity of,  248. 

Liter,  6. 

Local  action,  379. 

Lodestone,  359. 


Longitudinal  loops,  194. 
Longitudinal  waves,  171. 
Loudness  of  sound,  173. 

Machines,  efficiency  of,  99;  elec- 
trical, 354;  general  law  of,  84; 
simple,  83. 

Magdeburg  hemispheres,  147. 

Magnet,  artificial,  361 ;  electro-, 
391;  horseshoe,  361;  natural, 
359;  poles  of,  360. 

Magnetic  attraction,  361. 

Magnetic  field,  363. 

Magnetic  induction,  362. 

Magnetic  lines  of  force,  363. 

Magnetic  needle,  368. 

Magnetic  repulsion,   361. 

Magnetic  substances,  361. 

Magnetism,  induced,  362;  terres- 
trial, 368;  theory  of,  366. 

Magnifier,  simple,  311. 

Major  chord,   182. 

Major  scale,   181. 

Mass,  3;  center  of,  65;  measure- 
ment of,  9;  unit  of,  7. 

Matter,  2;  indestructibility  ol, 
3;  states  of,  11. 

Maxwell,  sketch  and  portrait  of, 
facing  432. 

Mechanical  advantage,  85. 

Mechanical  equivalent  of  heat, 
258. 

Melting  point,  231;  laws  of,  232. 

Meter,  standard,  4. 

Metric  system,  4. 

Microphone,  see  Transmitter. 

Microscope,  315. 

Mirrors,  278;  concave,  284;  con- 
vex, 285;  images  by,  281,  286, 
287;  principal  focus  of,  284. 

Mixtures,  method  of,  227. 

Molecular  energy,  211. 

Molecular  forces,  130. 

Molecular  theory  of  heat,  211. 

Molecules,  130. 

Moment  of  force,  89. 

Momentum,  28. 


478 


A   HIGH  SCHOOL  COURSE   IN   PHYSICS 


Mont   Blanc,   153. 

Morse,  453. 

Motion,  13;  accelerated,  17;  cir- 
cular, 44 ;  compounded,  24,  29 ; 
first  law  of,  28;  perpetual, 
259;  rectilinear,  13;  second  law 
of,  30;  third  law  of,  33;  uni- 
form, 13. 

Motor,  electric,  442;  water,  115. 

Musical  instruments,   197,  202. 

Musical  scale,  180. 

Needle,   dipping,   371;    magnetic, 

368. 
Newton,  sketch  and  portrait  of, 

facing  30. 

Newton's  law  of  gravitation,  62. 
Newton's    lawa    of    motion,    28, 

30,  33. 
Nodes,  in  pipes,  201;  in  strings, 

194. 

Noise,  179. 
Non-conductors,  339. 

Octave,  182. 

Oersted,  sketch  and  portrait  of, 

facing  382. 
Ohm,    sketch    and    portrait    of, 

facing  406;  unit,  406. 
Ohm's  law,  408. 
Opaque  bodies,  271. 
Opera  glass,  316. 
Optical  center,  308. 
Optical  instruments,  311. 
Organ  pipes,  open,  199;  overtones 

in,  201;   stopped,  199. 
Oscillation,  center  of,  79. 
Oscillatory  discharge,  462. 
Overtones,  194,  201. 

Parallel  connection  of  cells,  411. 
Parallel  forces,  40. 
Parallelogram  of  forces,  37. 
Partial  tones,  194. 
Pascal,   112,   141. 
Pascal's  law,   111. 
Pendulum,    compound,    78;    sim- 
ple, 74. 


Percussion,  center  of,  79. 

Period  of  vibration,  74. 

Permeability,  3G4. 

Phenomenon,  1. 

Phonograph,  204. 

Photoweter,  Bunsen's,  276. 

Physics,  definition  of,  1. 

Pigments,  color  of,  324;  mixing, 
324. 

Pisa,  tower  of,  68,  69. 

Pitch,  179;  of  pipes,  109;  of 
strings,  192;  wave  length  and, 
173. 

Points,  effect  of,  344. 

Polarization  of  cells,  383. 

Poles,  magnetic,  360. 

Potential,  difference  of,  347;  fall 
of,  381;  zero,  347. 

Potential  energy,  55. 

Power,  53;  candle,  276;  electri- 
cal, 414. 

Pressure,  in  liquids,  1Q3-;  of  com- 
pressed air,  160;  of  gases,  148; 
of  saturated  vapor,  238;  at- 
mospheric, 139. 

Primary  coil,  429. 

Principle  of  Archimedes,  119. 

Prism,  dispersion  by,  321;  re- 
fraction by,  298. 

Pulley,  85. 

Pump,  air,  155;  common  lift, 
156;  condensing,  156;  force, 
158. 

Quality  of  sounds,  196. 

Radiant  energy,  253;  absorption 
of,  253. 

Radiation,  252;   electrical,  463. 

Radio-activity,  468. 

Radiometer,  255. 

Radium,  468. 

Rainbow,  325. 

Ray,  of  light,  281;  alpha,  469; 
beta,  469;  cathode,  467;  gam- 
ma, 469. 

Receiver,'  telephone,  457,  458. 


INDEX 


479 


Reeds,  204. 

Reflection,     of     light,     278;     of 

sound,     176;     angle    of,    279; 

double,  282 ;  law  of,  278 ;  total, 

296. 
Refraction,  292;  explained,  293; 

index  of,  294;   laws^pf,  293. 
Regelation,  233. 
Relay,  455. 
Resistance,  of  batteries,  409;   of 

wires,     417;      electrical,     409; 

laws  of,  405;  measurement  of, 

418;   unit  of,  406. 
Resistance  coils,  418. 
Resolution  of  forces,  42. 
Resonance,  185;  explained,  187. 
Resultant,  21,   38. 
Retentivity,  362. 
Reversibility  of  pendulum,  79. 
Roemer,  269. 
Rontgen,  sketch  and  portrait  of, 

facing  466. 
Rontgen  rays,  467. 
Rowland,  258. 
Rumford,  sketch  and  portrait  of, 

facing  256. 
Rutherford,  471. 

Sailboat,  43. 

Saturation,  of  vapors,  238. 

Saturation  pressure,  238;    mag- 
netic, 267. 

Scale,  diatonic,  181 ;  musical,  180. 

Screw,  97;  mechanical  advantage 
of,  98. 

Secondary  coil,  429. 

Seconds  pendulum,  78. 

Series  connections,  410,  420. 

Shadows,  271. 

Sharps  and  flats,  183. 

Shunts,  421. 

Siphon,   159. 
.    Siren,  180. 

Soap  films,   130;  color  in,  332. 

Soddy,  471. 

Solar  spectrum,  324 ;  elements  of, 
329. 


Solenoid,  390. 

Solids,  density  of,  124;  sound  in, 

168. 

Sonometer  192. 
Sound,  166;  air  as  a  medium  of, 

167  r>  cause  of,    165;    intensity 

of,    173$   interference  of,    188; 

musicafc  179;  quality  of,  196; 

reflecti&i    of,    176;    speed    of, 

168;   wi-ve  motion  of,  169. 
Sounder,  ilelegraph,  453. 
Sounding  boards,  176. 
Spark,    electric,    352,    356,    430, 

463. 

Specific  gravity,  126. 
Specific  heat,  226;  measurement 

of,  227. 

Spectacles,  313. 
Spectra,    324;    bright-line,    328; 

conti|ajDUS,  327 ;  elements  iden- 

tifiecnSy,  329. 
Speed,   of  light,   269;    of  sound, 

168. 

Stability,  67. 
Stable  equilibrium,  66. 
Steam  engine,  260. 
Steam  turbine,  263. 
Steelyard,  93. 
Stereoscope,  317. 
Storage  cell,  399. 
Strength,  current,  402. 
Strings,  laws  of,  192,  193. 
Substance,  3. 
Suction  pump,  157. 
Sun,  as  source  of  energy,  255. 
Surface  tension,  132. 
Suspension,  point  of,  79. 
Sympathetic  vibrations,  185. 

Telegraph,  453;  wireless,  464. 

Telephone,  electric,  457;  mechan- 
ical, 201. 

Telescope,  315;  Galileo's,  316. 

Temperament,  184. 

Temperature,  208 ;  absolute,  222 ; 
low,  223 ;  measurement  of,  212 ; 
scales  of,  213. 


480 


A  HIGH  SCHOOL  COURSE  IN   PHYSICS 


Tempered  scales,  184. 

Tempering,  184. 

Thermal  capacity,  226. 

Thermometer,  centigrade,  213; 
comparison  of,  214;  dial,  218; 
Fahrenheit,  214;  fixed  points 
of,  212;  Galileo's  air,  215; 

*  graduation  of,  213;  range  of, 
215. 

Thunder,  345. 

Time,  unit  of,  11. 

Toeppler-Holtz  electrical  ma- 
chine, 354. 

Torricelli's  experiment,  141. 

Transformation  of  energy,  58. 

Transformer,  445. 

Transmission,  of  electricity,  415; 
of  heat,  247;  of  light,  268;  of 
sound,  167. 

Transmitter,  458. 

Transparent  bodies,  color  of,  323. 

Transverse  waves,  170. 

Tuning  fork,  185. 

Turbine,  water,  116;  steam,  263. 

Units,  of  area,  6;  of  candle  pow- 
er, 276 ;  of  capacity,  6 ;  of  cur- 
rent, 402;  of  electricity,  348; 
of  force,  30;  of  heat,  225;  of 
length,  4;  of  mass,  7;  of  po- 
tential, 407;  of  power,  53;  of 
resistance,  406;  of  time,  11; 
of  volume,  6;  of  work,  52. 

Vacuum,  Torricellian,  142. 

Vapor,  237;   saturated,  238. 

Vapor  pressure,   238. 

Vaporization,  243;  heat  of,  249. 

Vector,  16. 

Velocity,  14;  at  any  instant,  15; 
of  falling  bodies,  71;  of  light, 
269;  of  sound,  168;  average, 
15;  composition  of,  22,  24; 
representation  of,  15. 


Ventilation,  451.     . 

Vibrating  body,  165. 

Vibration,     of     pipes,     200;     of 

strings,  192;  sympathetic,  185. 
Vibration  numbers,   181,  184. 
Volt,  407. 
Volta,  375;    sketch  and  portrait 

of,  facing  352. 
Voltaic  cell,  375. 
Voltmeter,  407. 

Water,  abnormal  expansion  of, 
220;  city  supply  of,  114;  den- 
sity of,  10J.;  greatest  density 
of,  221. 

Water  wheels,  115. 

Water  turbine,  116. 

Watt,  54,  415. 

Watt-hour,  449. 

Wave  length,  173;  of  light,  322; 
equation  of,  173. 

Waves,  electrical,  463;  kinds  of, 
170;  longitudinal  and  trans- 
verse, 170;  sound,  171;  trans- 
mission of,  171. 

Wedge,  98. 

Weighing  devices,  10. 

Weight,  263;  and  mass,  9;  law 
of,  63. 

Wheatstone's  bridge,  419. 

Wheel  and  axle,  93;  mechanical 
advantage  of,  94. 

White  light,  321. 

Windlass,  94. 

Wireless  telegraphy,  464. 

Work,  51;  principle  of,  83;  rate 
of,  53;  units  of,  52. 

X-rays,  467;  properties  of,  468. 
Yard,  metric  equivalent  of,  5. 
Zero  potential,  347. 


(2) 


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